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|00000050| 6f 2e 20 20 54 68 65 20 | 65 71 75 61 74 69 6f 6e |o.  The |equation|
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|00000070| 20 74 79 70 65 20 77 69 | 74 68 20 11 33 75 20 11 | type wi|th .3u .|
|00000080| 31 3d 20 11 33 78 20 20 | 20 11 31 2e 20 20 57 65 |1= .3x  | .1.  We|
|00000090| 20 73 6f 6c 76 65 20 61 | 73 20 66 6f 6c 6c 6f 77 | solve a|s follow|
|000000a0| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 11 32 |s..0... |      .2|
|000000b0| 32 0d 0b 00 20 20 20 20 | 20 11 31 33 11 33 75 20 |2...    | .13.3u |
|000000c0| 20 11 31 2b 20 38 11 33 | 75 20 11 31 2d 20 33 20 | .1+ 8.3|u .1- 3 |
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|000000e0| 2d 20 31 29 28 11 33 75 | 20 11 31 2b 20 33 29 20 |- 1)(.3u| .1+ 3) |
|000000f0| 3d 20 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |= 0...  |        |
|00000100| 20 33 11 33 75 20 11 31 | 2d 20 31 20 3d 20 30 20 | 3.3u .1|- 1 = 0 |
|00000110| 20 20 11 34 35 35 36 20 | 20 20 11 33 75 20 11 31 |  .4556 |  .3u .1|
|00000120| 3d 20 31 2f 33 0d 0a 00 | 20 20 20 20 20 20 20 20 |= 1/3...|        |
|00000130| 20 20 20 20 11 33 75 20 | 11 31 2b 20 33 20 3d 20 |    .3u |.1+ 3 = |
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|000001c0| 11 31 3d 20 31 2f 33 20 | 20 20 20 20 20 20 61 6e |.1= 1/3 |      an|
|000001d0| 64 20 20 20 20 20 20 20 | 11 33 75 20 11 31 3d 20 |d       |.3u .1= |
|000001e0| 2d 33 0d 0a 00 20 20 20 | 11 32 31 2f 33 20 20 20 |-3...   |.21/3   |
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|00000200| 20 20 31 2f 33 0d 0b 00 | 20 20 11 33 78 20 20 20 |  1/3...|  .3x   |
|00000210| 20 11 31 3d 20 31 2f 33 | 20 20 20 20 20 20 20 20 | .1= 1/3|        |
|00000220| 20 20 20 20 20 20 11 33 | 78 20 20 20 20 11 31 3d |      .3|x    .1=|
|00000230| 20 2d 33 0d 0a 00 0d 0b | 00 20 20 20 20 20 11 33 | -3.....|.     .3|
|00000240| 78 20 11 31 3d 20 31 2f | 32 37 20 20 20 20 20 20 |x .1= 1/|27      |
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|00000260| 3d 20 2d 32 37 0d 0a 00 | 00 17 4f 0d 20 20 20 20 |= -27...|..O.    |
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|00000290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 31 |        |     .21|
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|000002d0| 74 68 20 11 33 75 20 11 | 31 3d 20 11 33 78 20 20 |th .3u .|1= .3x  |
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|000002f0| 73 20 66 6f 6c 6c 6f 77 | 73 2e 12 30 0d 0a 00 20 |s follow|s..0... |
|00000300| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 20 20 20 20 |      .2|2...    |
|00000310| 20 11 31 33 11 33 75 20 | 20 11 31 2b 20 38 11 33 | .13.3u | .1+ 8.3|
|00000320| 75 20 11 31 2d 20 33 20 | 3d 20 30 0d 0a 00 20 20 |u .1- 3 |= 0...  |
|00000330| 28 33 11 33 75 20 11 31 | 2d 20 31 29 28 11 33 75 |(3.3u .1|- 1)(.3u|
|00000340| 20 11 31 2b 20 33 29 20 | 3d 20 30 0d 0a 00 20 20 | .1+ 3) |= 0...  |
|00000350| 20 20 20 20 20 20 20 20 | 20 33 11 33 75 20 11 31 |        | 3.3u .1|
|00000360| 2d 20 31 20 3d 20 30 20 | 20 20 11 34 35 35 36 20 |- 1 = 0 |  .4556 |
|00000370| 20 20 11 33 75 20 11 31 | 3d 20 31 2f 33 0d 0a 00 |  .3u .1|= 1/3...|
|00000380| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 33 75 20 |        |    .3u |
|00000390| 11 31 2b 20 33 20 3d 20 | 30 20 20 20 11 34 35 35 |.1+ 3 = |0   .455|
|000003a0| 36 20 20 20 11 33 75 20 | 11 31 3d 20 2d 33 0d 0a |6   .3u |.1= -3..|
|000003b0| 00 0d 0b 00 12 33 55 73 | 69 6e 67 20 74 68 65 20 |.....3Us|ing the |
|000003c0| 11 33 75 11 31 2d 73 6f | 6c 75 74 69 6f 6e 73 20 |.3u.1-so|lutions |
|000003d0| 6f 66 20 31 2f 33 20 61 | 6e 64 20 2d 33 2c 20 77 |of 1/3 a|nd -3, w|
|000003e0| 65 20 6f 62 74 61 69 6e | 20 74 68 65 20 66 6f 6c |e obtain| the fol|
|000003f0| 6c 6f 77 69 6e 67 20 11 | 33 78 11 31 2d 73 6f 6c |lowing .|3x.1-sol|
|00000400| 75 74 69 6f 6e 73 2e 12 | 30 0d 0a 00 0d 0b 00 20 |utions..|0...... |
|00000410| 20 20 20 20 11 33 75 20 | 11 31 3d 20 31 2f 33 20 |    .3u |.1= 1/3 |
|00000420| 20 20 20 20 20 20 61 6e | 64 20 20 20 20 20 20 20 |      an|d       |
|00000430| 11 33 75 20 11 31 3d 20 | 2d 33 0d 0a 00 20 20 20 |.3u .1= |-3...   |
|00000440| 11 32 31 2f 33 20 20 20 | 20 20 20 20 20 20 20 20 |.21/3   |        |
|00000450| 20 20 20 20 20 20 20 20 | 20 20 31 2f 33 0d 0b 00 |        |  1/3...|
|00000460| 20 20 11 33 78 20 20 20 | 20 11 31 3d 20 31 2f 33 |  .3x   | .1= 1/3|
|00000470| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |        |      .3|
|00000480| 78 20 20 20 20 11 31 3d | 20 2d 33 0d 0a 00 0d 0b |x    .1=| -3.....|
|00000490| 00 20 20 20 20 20 11 33 | 78 20 11 31 3d 20 31 2f |.     .3|x .1= 1/|
|000004a0| 32 37 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |27      |        |
|000004b0| 20 20 11 33 78 20 11 31 | 3d 20 2d 32 37 0d 0a 00 |  .3x .1|= -27...|
|000004c0| 00 17 4f 0d 20 20 20 20 | 20 20 20 20 20 20 20 20 |..O.    |        |
|000004d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000004e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000004f0| 20 20 20 20 20 11 32 31 | 2f 33 0d 0b 00 11 31 12 |     .21|/3....1.|
|00000500| 33 4e 6f 2e 20 20 54 68 | 65 20 65 71 75 61 74 69 |3No.  Th|e equati|
|00000510| 6f 6e 20 69 73 20 6f 66 | 20 71 75 61 64 72 61 74 |on is of| quadrat|
|00000520| 69 63 20 74 79 70 65 20 | 77 69 74 68 20 11 33 75 |ic type |with .3u|
|00000530| 20 11 31 3d 20 11 33 78 | 20 20 20 11 31 2e 20 20 | .1= .3x|   .1.  |
|00000540| 57 65 20 73 6f 6c 76 65 | 20 61 73 20 66 6f 6c 6c |We solve| as foll|
|00000550| 6f 77 73 2e 12 30 0d 0a | 00 20 20 20 20 20 20 20 |ows..0..|.       |
|00000560| 11 32 32 0d 0b 00 20 20 | 20 20 20 11 31 33 11 33 |.22...  |   .13.3|
|00000570| 75 20 20 11 31 2b 20 38 | 11 33 75 20 11 31 2d 20 |u  .1+ 8|.3u .1- |
|00000580| 33 20 3d 20 30 0d 0a 00 | 20 20 28 33 11 33 75 20 |3 = 0...|  (3.3u |
|00000590| 11 31 2d 20 31 29 28 11 | 33 75 20 11 31 2b 20 33 |.1- 1)(.|3u .1+ 3|
|000005a0| 29 20 3d 20 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |) = 0...|        |
|000005b0| 20 20 20 33 11 33 75 20 | 11 31 2d 20 31 20 3d 20 |   3.3u |.1- 1 = |
|000005c0| 30 20 20 20 11 34 35 35 | 36 20 20 20 11 33 75 20 |0   .455|6   .3u |
|000005d0| 11 31 3d 20 31 2f 33 0d | 0a 00 20 20 20 20 20 20 |.1= 1/3.|..      |
|000005e0| 20 20 20 20 20 20 11 33 | 75 20 11 31 2b 20 33 20 |      .3|u .1+ 3 |
|000005f0| 3d 20 30 20 20 20 11 34 | 35 35 36 20 20 20 11 33 |= 0   .4|556   .3|
|00000600| 75 20 11 31 3d 20 2d 33 | 0d 0a 00 0d 0b 00 12 33 |u .1= -3|.......3|
|00000610| 55 73 69 6e 67 20 74 68 | 65 20 11 33 75 11 31 2d |Using th|e .3u.1-|
|00000620| 73 6f 6c 75 74 69 6f 6e | 73 20 6f 66 20 31 2f 33 |solution|s of 1/3|
|00000630| 20 61 6e 64 20 2d 33 2c | 20 77 65 20 6f 62 74 61 | and -3,| we obta|
|00000640| 69 6e 20 74 68 65 20 66 | 6f 6c 6c 6f 77 69 6e 67 |in the f|ollowing|
|00000650| 20 11 33 78 11 31 2d 73 | 6f 6c 75 74 69 6f 6e 73 | .3x.1-s|olutions|
|00000660| 2e 12 30 0d 0a 00 0d 0b | 00 20 20 20 20 20 11 33 |..0.....|.     .3|
|00000670| 75 20 11 31 3d 20 31 2f | 33 20 20 20 20 20 20 20 |u .1= 1/|3       |
|00000680| 61 6e 64 20 20 20 20 20 | 20 20 11 33 75 20 11 31 |and     |  .3u .1|
|00000690| 3d 20 2d 33 0d 0a 00 20 | 20 20 11 32 31 2f 33 20 |= -3... |  .21/3 |
|000006a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000006b0| 20 20 20 20 31 2f 33 0d | 0b 00 20 20 11 33 78 20 |    1/3.|..  .3x |
|000006c0| 20 20 20 11 31 3d 20 31 | 2f 33 20 20 20 20 20 20 |   .1= 1|/3      |
|000006d0| 20 20 20 20 20 20 20 20 | 11 33 78 20 20 20 20 11 |        |.3x    .|
|000006e0| 31 3d 20 2d 33 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |1= -3...|...     |
|000006f0| 11 33 78 20 11 31 3d 20 | 31 2f 32 37 20 20 20 20 |.3x .1= |1/27    |
|00000700| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 33 78 20 |        |    .3x |
|00000710| 11 31 3d 20 2d 32 37 0d | 0a 00 00 17 4f 03 0d 0b |.1= -27.|....O...|
|00000720| 00 52 69 67 68 74 2e 20 | 20 54 68 65 20 67 69 76 |.Right. | The giv|
|00000730| 65 6e 20 65 71 75 61 74 | 69 6f 6e 20 68 61 73 20 |en equat|ion has |
|00000740| 74 77 6f 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f |two real| solutio|
|00000750| 6e 73 3a 20 31 2f 32 37 | 20 61 6e 64 20 2d 32 37 |ns: 1/27| and -27|
|00000760| 2e 0d 0a 00 00 17 4f 0d | 0d 0b 00 12 33 57 72 6f |......O.|....3Wro|
|00000770| 6e 67 2e 20 20 57 65 20 | 73 6f 6c 76 65 20 74 68 |ng.  We |solve th|
|00000780| 65 20 65 71 75 61 74 69 | 6f 6e 20 61 73 20 66 6f |e equati|on as fo|
|00000790| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 20 20 20 20 11 |llows..0|...    .|
|000007a0| 34 44 32 32 32 32 32 32 | 0d 0b 00 20 20 20 53 20 |4D222222|...   S |
|000007b0| 11 31 33 11 33 79 20 11 | 31 2b 20 35 20 2d 20 33 |.13.3y .|1+ 5 - 3|
|000007c0| 20 3d 20 34 20 20 20 20 | 20 20 20 20 20 20 20 11 | = 4    |       .|
|000007d0| 32 12 31 47 69 76 65 6e | 20 65 71 75 61 74 69 6f |2.1Given| equatio|
|000007e0| 6e 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 11 34 |n.0...  |      .4|
|000007f0| 44 32 32 32 32 32 32 0d | 0b 00 20 20 20 20 20 20 |D222222.|..      |
|00000800| 20 53 20 11 31 33 11 33 | 79 20 11 31 2b 20 35 20 | S .13.3|y .1+ 5 |
|00000810| 3d 20 37 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |= 7     |      .2|
|00000820| 12 31 41 64 64 20 33 20 | 74 6f 20 62 6f 74 68 20 |.1Add 3 |to both |
|00000830| 73 69 64 65 73 12 30 0d | 0a 00 0d 0b 00 20 20 20 |sides.0.|.....   |
|00000840| 20 20 20 20 20 20 11 31 | 33 11 33 79 20 11 31 2b |      .1|3.3y .1+|
|00000850| 20 35 20 3d 20 34 39 20 | 20 20 20 20 20 20 20 20 | 5 = 49 |        |
|00000860| 20 11 32 12 31 53 71 75 | 61 72 65 20 62 6f 74 68 | .2.1Squ|are both|
|00000870| 20 73 69 64 65 73 12 30 | 0d 0a 00 0d 0b 00 20 20 | sides.0|......  |
|00000880| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 33 11 33 |        |   .13.3|
|00000890| 79 20 11 31 3d 20 34 34 | 20 20 20 20 20 20 20 20 |y .1= 44|        |
|000008a0| 20 20 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |  ......|        |
|000008b0| 20 20 20 20 20 20 11 33 | 79 20 11 31 3d 20 34 34 |      .3|y .1= 44|
|000008c0| 2f 33 20 20 20 20 20 20 | 20 20 11 32 12 31 54 72 |/3      |  .2.1Tr|
|000008d0| 69 61 6c 20 73 6f 6c 75 | 74 69 6f 6e 12 33 0d 0a |ial solu|tion.3..|
|000008e0| 00 0d 0b 00 11 31 41 66 | 74 65 72 20 63 68 65 63 |.....1Af|ter chec|
|000008f0| 6b 69 6e 67 20 74 68 69 | 73 20 73 6f 6c 75 74 69 |king thi|s soluti|
|00000900| 6f 6e 2c 20 77 65 20 63 | 61 6e 20 63 6f 6e 63 6c |on, we c|an concl|
|00000910| 75 64 65 20 74 68 61 74 | 20 74 68 65 20 73 6f 6c |ude that| the sol|
|00000920| 75 74 69 6f 6e 20 69 73 | 20 11 33 79 20 11 31 3d |ution is| .3y .1=|
|00000930| 20 34 34 2f 33 2e 0d 0a | 00 28 57 65 20 6c 65 61 | 44/3...|.(We lea|
|00000940| 76 65 20 74 68 65 20 63 | 68 65 63 6b 69 6e 67 20 |ve the c|hecking |
|00000950| 74 6f 20 79 6f 75 2e 29 | 0d 0a 00 00 17 4f 03 0d |to you.)|.....O..|
|00000960| 0b 00 52 69 67 68 74 2e | 20 20 54 68 65 20 73 6f |..Right.|  The so|
|00000970| 6c 75 74 69 6f 6e 20 74 | 6f 20 74 68 65 20 67 69 |lution t|o the gi|
|00000980| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 20 69 73 20 |ven equa|tion is |
|00000990| 34 34 2f 33 2e 0d 0a 00 | 00 17 4f 0d 0d 0b 00 12 |44/3....|..O.....|
|000009a0| 33 57 72 6f 6e 67 2e 20 | 20 57 65 20 73 6f 6c 76 |3Wrong. | We solv|
|000009b0| 65 20 74 68 65 20 65 71 | 75 61 74 69 6f 6e 20 61 |e the eq|uation a|
|000009c0| 73 20 66 6f 6c 6c 6f 77 | 73 2e 12 30 0d 0a 00 20 |s follow|s..0... |
|000009d0| 20 20 20 11 34 44 32 32 | 32 32 32 32 0d 0b 00 20 |   .4D22|2222... |
|000009e0| 20 20 53 20 11 31 33 11 | 33 79 20 11 31 2b 20 35 |  S .13.|3y .1+ 5|
|000009f0| 20 2d 20 33 20 3d 20 34 | 20 20 20 20 20 20 20 20 | - 3 = 4|        |
|00000a00| 20 20 20 11 32 12 31 47 | 69 76 65 6e 20 65 71 75 |   .2.1G|iven equ|
|00000a10| 61 74 69 6f 6e 12 30 0d | 0a 00 20 20 20 20 20 20 |ation.0.|..      |
|00000a20| 20 20 11 34 44 32 32 32 | 32 32 32 0d 0b 00 20 20 |  .4D222|222...  |
|00000a30| 20 20 20 20 20 53 20 11 | 31 33 11 33 79 20 11 31 |     S .|13.3y .1|
|00000a40| 2b 20 35 20 3d 20 37 20 | 20 20 20 20 20 20 20 20 |+ 5 = 7 |        |
|00000a50| 20 20 11 32 12 31 41 64 | 64 20 33 20 74 6f 20 62 |  .2.1Ad|d 3 to b|
|00000a60| 6f 74 68 20 73 69 64 65 | 73 12 30 0d 0a 00 0d 0b |oth side|s.0.....|
|00000a70| 00 20 20 20 20 20 20 20 | 20 20 11 31 33 11 33 79 |.       |  .13.3y|
|00000a80| 20 11 31 2b 20 35 20 3d | 20 34 39 20 20 20 20 20 | .1+ 5 =| 49     |
|00000a90| 20 20 20 20 20 11 32 12 | 31 53 71 75 61 72 65 20 |     .2.|1Square |
|00000aa0| 62 6f 74 68 20 73 69 64 | 65 73 12 30 0d 0a 00 0d |both sid|es.0....|
|00000ab0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |..      |       .|
|00000ac0| 31 33 11 33 79 20 11 31 | 3d 20 34 34 20 20 20 20 |13.3y .1|= 44    |
|00000ad0| 20 20 20 20 20 20 0d 0a | 00 0d 0b 00 20 20 20 20 |      ..|....    |
|00000ae0| 20 20 20 20 20 20 20 20 | 20 20 11 33 79 20 11 31 |        |  .3y .1|
|00000af0| 3d 20 34 34 2f 33 20 20 | 20 20 20 20 20 20 11 32 |= 44/3  |      .2|
|00000b00| 12 31 54 72 69 61 6c 20 | 73 6f 6c 75 74 69 6f 6e |.1Trial |solution|
|00000b10| 12 33 0d 0a 00 0d 0b 00 | 11 31 41 66 74 65 72 20 |.3......|.1After |
|00000b20| 63 68 65 63 6b 69 6e 67 | 20 74 68 69 73 20 73 6f |checking| this so|
|00000b30| 6c 75 74 69 6f 6e 2c 20 | 77 65 20 63 61 6e 20 63 |lution, |we can c|
|00000b40| 6f 6e 63 6c 75 64 65 20 | 74 68 61 74 20 74 68 65 |onclude |that the|
|00000b50| 20 73 6f 6c 75 74 69 6f | 6e 20 69 73 20 11 33 79 | solutio|n is .3y|
|00000b60| 20 11 31 3d 20 34 34 2f | 33 2e 0d 0a 00 28 57 65 | .1= 44/|3....(We|
|00000b70| 20 6c 65 61 76 65 20 74 | 68 65 20 63 68 65 63 6b | leave t|he check|
|00000b80| 69 6e 67 20 74 6f 20 79 | 6f 75 2e 29 0d 0a 00 00 |ing to y|ou.)....|
|00000b90| 17 4f 03 0d 0b 00 57 72 | 6f 6e 67 2e 20 20 54 68 |.O....Wr|ong.  Th|
|00000ba0| 65 20 65 71 75 61 74 69 | 6f 6e 20 64 6f 65 73 20 |e equati|on does |
|00000bb0| 68 61 76 65 20 61 20 73 | 6f 6c 75 74 69 6f 6e 2e |have a s|olution.|
|00000bc0| 20 20 54 72 79 20 61 67 | 61 69 6e 2e 0d 0a 00 00 |  Try ag|ain.....|
|00000bd0| 17 4f 0b 0d 0b 00 12 33 | 57 72 6f 6e 67 2e 20 20 |.O.....3|Wrong.  |
|00000be0| 57 65 20 73 6f 6c 76 65 | 20 61 73 20 66 6f 6c 6c |We solve| as foll|
|00000bf0| 6f 77 73 2e 12 30 0d 0a | 00 20 20 20 20 11 34 44 |ows..0..|.    .4D|
|00000c00| 32 32 32 32 32 32 0d 0b | 00 20 20 20 53 20 11 31 |222222..|.   S .1|
|00000c10| 38 11 33 78 20 11 31 2b | 20 31 20 3d 20 11 33 78 |8.3x .1+| 1 = .3x|
|00000c20| 20 11 31 2b 20 32 20 20 | 20 20 20 20 20 20 20 20 | .1+ 2  |        |
|00000c30| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00000c40| 53 75 62 74 72 61 63 74 | 20 32 20 66 72 6f 6d 20 |Subtract| 2 from |
|00000c50| 62 6f 74 68 20 73 69 64 | 65 73 12 30 0d 0a 00 20 |both sid|es.0... |
|00000c60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 32 0d |        |      2.|
|00000c70| 0b 00 20 20 20 20 20 11 | 31 38 11 33 78 20 11 31 |..     .|18.3x .1|
|00000c80| 2b 20 31 20 3d 20 11 33 | 78 20 20 11 31 2b 20 34 |+ 1 = .3|x  .1+ 4|
|00000c90| 11 33 78 20 11 31 2b 20 | 34 20 20 20 20 20 20 20 |.3x .1+ |4       |
|00000ca0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 53 71 75 |        | .2.1Squ|
|00000cb0| 61 72 65 20 62 6f 74 68 | 20 73 69 64 65 73 12 30 |are both| sides.0|
|00000cc0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00000cd0| 20 20 32 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |  2...  |        |
|00000ce0| 11 31 30 20 3d 20 11 33 | 78 20 20 11 31 2d 20 34 |.10 = .3|x  .1- 4|
|00000cf0| 11 33 78 20 11 31 2b 20 | 33 20 20 20 20 20 20 20 |.3x .1+ |3       |
|00000d00| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 53 74 61 |        | .2.1Sta|
|00000d10| 6e 64 61 72 64 20 66 6f | 72 6d 12 30 0d 0a 00 0d |ndard fo|rm.0....|
|00000d20| 0b 00 20 20 20 20 20 20 | 20 20 20 20 11 31 30 20 |..      |    .10 |
|00000d30| 3d 20 28 11 33 78 20 11 | 31 2d 20 33 29 28 11 33 |= (.3x .|1- 3)(.3|
|00000d40| 78 20 11 31 2d 20 31 29 | 20 20 20 20 20 20 20 20 |x .1- 1)|        |
|00000d50| 20 20 20 20 20 11 32 12 | 31 46 61 63 74 6f 72 65 |     .2.|1Factore|
|00000d60| 64 20 66 6f 72 6d 12 30 | 0d 0a 00 0d 0b 00 20 20 |d form.0|......  |
|00000d70| 20 20 20 20 20 20 20 20 | 11 31 30 20 3d 20 11 33 |        |.10 = .3|
|00000d80| 78 20 11 31 2d 20 33 20 | 20 20 11 34 35 35 36 20 |x .1- 3 |  .4556 |
|00000d90| 20 20 11 31 33 20 3d 20 | 11 33 78 20 20 20 20 20 |  .13 = |.3x     |
|00000da0| 20 20 20 11 32 12 31 53 | 65 74 20 66 69 72 73 74 |   .2.1S|et first|
|00000db0| 20 66 61 63 74 6f 72 20 | 65 71 75 61 6c 20 74 6f | factor |equal to|
|00000dc0| 20 30 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 | 0.0... |        |
|00000dd0| 20 11 31 30 20 3d 20 11 | 33 78 20 11 31 2d 20 31 | .10 = .|3x .1- 1|
|00000de0| 20 20 20 11 34 35 35 36 | 20 20 20 11 31 31 20 3d |   .4556|   .11 =|
|00000df0| 20 11 33 78 20 20 20 20 | 20 20 20 20 11 32 12 31 | .3x    |    .2.1|
|00000e00| 53 65 74 20 73 65 63 6f | 6e 64 20 66 61 63 74 6f |Set seco|nd facto|
|00000e10| 72 20 65 71 75 61 6c 20 | 74 6f 20 30 12 30 0d 0a |r equal |to 0.0..|
|00000e20| 00 00 17 4f 0b 0d 0b 00 | 12 33 57 72 6f 6e 67 2e |...O....|.3Wrong.|
|00000e30| 20 20 57 65 20 73 6f 6c | 76 65 20 61 73 20 66 6f |  We sol|ve as fo|
|00000e40| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 20 20 20 20 11 |llows..0|...    .|
|00000e50| 34 44 32 32 32 32 32 32 | 0d 0b 00 20 20 20 53 20 |4D222222|...   S |
|00000e60| 11 31 38 11 33 78 20 11 | 31 2b 20 31 20 3d 20 11 |.18.3x .|1+ 1 = .|
|00000e70| 33 78 20 11 31 2b 20 32 | 20 20 20 20 20 20 20 20 |3x .1+ 2|        |
|00000e80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00000e90| 12 31 53 75 62 74 72 61 | 63 74 20 32 20 66 72 6f |.1Subtra|ct 2 fro|
|00000ea0| 6d 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0a |m both s|ides.0..|
|00000eb0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00000ec0| 32 0d 0b 00 20 20 20 20 | 20 11 31 38 11 33 78 20 |2...    | .18.3x |
|00000ed0| 11 31 2b 20 31 20 3d 20 | 11 33 78 20 20 11 31 2b |.1+ 1 = |.3x  .1+|
|00000ee0| 20 34 11 33 78 20 11 31 | 2b 20 34 20 20 20 20 20 | 4.3x .1|+ 4     |
|00000ef0| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |        |   .2.1S|
|00000f00| 71 75 61 72 65 20 62 6f | 74 68 20 73 69 64 65 73 |quare bo|th sides|
|00000f10| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|00000f20| 20 20 20 20 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |    2...|        |
|00000f30| 20 20 11 31 30 20 3d 20 | 11 33 78 20 20 11 31 2d |  .10 = |.3x  .1-|
|00000f40| 20 34 11 33 78 20 11 31 | 2b 20 33 20 20 20 20 20 | 4.3x .1|+ 3     |
|00000f50| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |        |   .2.1S|
|00000f60| 74 61 6e 64 61 72 64 20 | 66 6f 72 6d 12 30 0d 0a |tandard |form.0..|
|00000f70| 00 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 11 31 |....    |      .1|
|00000f80| 30 20 3d 20 28 11 33 78 | 20 11 31 2d 20 33 29 28 |0 = (.3x| .1- 3)(|
|00000f90| 11 33 78 20 11 31 2d 20 | 31 29 20 20 20 20 20 20 |.3x .1- |1)      |
|00000fa0| 20 20 20 20 20 20 20 11 | 32 12 31 46 61 63 74 6f |       .|2.1Facto|
|00000fb0| 72 65 64 20 66 6f 72 6d | 12 30 0d 0a 00 0d 0b 00 |red form|.0......|
|00000fc0| 20 20 20 20 20 20 20 20 | 20 20 11 31 30 20 3d 20 |        |  .10 = |
|00000fd0| 11 33 78 20 11 31 2d 20 | 33 20 20 20 11 34 35 35 |.3x .1- |3   .455|
|00000fe0| 36 20 20 20 11 31 33 20 | 3d 20 11 33 78 20 20 20 |6   .13 |= .3x   |
|00000ff0| 20 20 20 20 20 11 32 12 | 31 53 65 74 20 66 69 72 |     .2.|1Set fir|
|00001000| 73 74 20 66 61 63 74 6f | 72 20 65 71 75 61 6c 20 |st facto|r equal |
|00001010| 74 6f 20 30 12 30 0d 0a | 00 20 20 20 20 20 20 20 |to 0.0..|.       |
|00001020| 20 20 20 11 31 30 20 3d | 20 11 33 78 20 11 31 2d |   .10 =| .3x .1-|
|00001030| 20 31 20 20 20 11 34 35 | 35 36 20 20 20 11 31 31 | 1   .45|56   .11|
|00001040| 20 3d 20 11 33 78 20 20 | 20 20 20 20 20 20 11 32 | = .3x  |      .2|
|00001050| 12 31 53 65 74 20 73 65 | 63 6f 6e 64 20 66 61 63 |.1Set se|cond fac|
|00001060| 74 6f 72 20 65 71 75 61 | 6c 20 74 6f 20 30 12 30 |tor equa|l to 0.0|
|00001070| 0d 0a 00 00 17 4f 03 0d | 0b 00 52 69 67 68 74 2e |.....O..|..Right.|
|00001080| 20 20 54 68 65 20 67 69 | 76 65 6e 20 65 71 75 61 |  The gi|ven equa|
|00001090| 74 69 6f 6e 20 68 61 73 | 20 74 77 6f 20 73 6f 6c |tion has| two sol|
|000010a0| 75 74 69 6f 6e 73 3a 20 | 31 20 61 6e 64 20 33 2e |utions: |1 and 3.|
|000010b0| 0d 0a 00 00 17 4f 03 0d | 0b 00 57 72 6f 6e 67 2e |.....O..|..Wrong.|
|000010c0| 20 20 54 68 69 73 20 65 | 71 75 61 74 69 6f 6e 20 |  This e|quation |
|000010d0| 68 61 73 20 74 77 6f 20 | 72 65 61 6c 20 73 6f 6c |has two |real sol|
|000010e0| 75 74 69 6f 6e 73 2e 20 | 20 54 72 79 20 61 67 61 |utions. | Try aga|
|000010f0| 69 6e 2e 0d 0a 00 00 17 | 4f 0c 0d 0b 00 12 33 57 |in......|O.....3W|
|00001100| 72 6f 6e 67 2e 20 20 57 | 65 20 73 6f 6c 76 65 20 |rong.  W|e solve |
|00001110| 74 68 65 20 65 71 75 61 | 74 69 6f 6e 20 61 73 20 |the equa|tion as |
|00001120| 66 6f 6c 6c 6f 77 73 2e | 12 30 0d 0a 00 20 20 20 |follows.|.0...   |
|00001130| 20 20 11 34 44 32 32 32 | 32 32 32 20 20 20 20 20 |  .4D222|222     |
|00001140| 20 20 20 44 32 32 0d 0b | 00 20 20 20 20 53 20 11 |   D22..|.    S .|
|00001150| 31 32 11 33 74 20 11 31 | 2b 20 33 20 3d 20 33 20 |12.3t .1|+ 3 = 3 |
|00001160| 2d 20 11 34 53 20 11 31 | 32 11 33 74 20 20 20 20 |- .4S .1|2.3t    |
|00001170| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 0d 0a |        |      ..|
|00001180| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001190| 20 20 20 20 20 20 11 34 | 44 32 32 0d 0b 00 20 20 |      .4|D22...  |
|000011a0| 20 20 20 20 11 31 32 11 | 33 74 20 11 31 2b 20 33 |    .12.|3t .1+ 3|
|000011b0| 20 3d 20 39 20 2d 20 36 | 11 34 53 20 11 31 32 11 | = 9 - 6|.4S .12.|
|000011c0| 33 74 20 11 31 2b 20 32 | 11 33 74 20 20 20 20 20 |3t .1+ 2|.3t     |
|000011d0| 20 20 20 20 20 20 20 11 | 32 12 31 53 71 75 61 72 |       .|2.1Squar|
|000011e0| 65 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0a |e both s|ides.0..|
|000011f0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001200| 20 20 20 11 34 44 32 32 | 0d 0b 00 20 20 20 20 20 |   .4D22|...     |
|00001210| 20 20 20 20 20 11 31 2d | 36 20 3d 20 2d 36 11 34 |     .1-|6 = -6.4|
|00001220| 53 20 11 31 32 11 33 74 | 20 20 20 20 20 20 20 20 |S .12.3t|        |
|00001230| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00001240| 49 73 6f 6c 61 74 65 20 | 72 61 64 69 63 61 6c 12 |Isolate |radical.|
|00001250| 30 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |0...    |        |
|00001260| 20 20 20 20 11 34 44 32 | 32 0d 0b 00 20 20 20 20 |    .4D2|2...    |
|00001270| 20 20 20 20 20 20 20 11 | 31 31 20 3d 20 11 34 53 |       .|11 = .4S|
|00001280| 20 11 31 32 11 33 74 20 | 20 20 20 20 20 20 20 20 | .12.3t |        |
|00001290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|000012a0| 31 44 69 76 69 64 65 20 | 62 6f 74 68 20 73 69 64 |1Divide |both sid|
|000012b0| 65 73 20 62 79 20 2d 36 | 12 30 0d 0a 00 0d 0b 00 |es by -6|.0......|
|000012c0| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 31 20 3d |        |   .11 =|
|000012d0| 20 32 11 33 74 20 20 20 | 20 20 20 20 20 20 20 20 | 2.3t   |        |
|000012e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|000012f0| 31 53 71 75 61 72 65 20 | 62 6f 74 68 20 73 69 64 |1Square |both sid|
|00001300| 65 73 12 30 0d 0a 00 0d | 0b 00 20 20 20 20 20 20 |es.0....|..      |
|00001310| 20 20 20 11 31 31 2f 32 | 20 3d 20 11 33 74 20 20 |   .11/2| = .3t  |
|00001320| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001330| 20 20 20 20 20 20 20 11 | 32 12 31 53 6f 6c 75 74 |       .|2.1Solut|
|00001340| 69 6f 6e 12 30 0d 0a 00 | 00 17 4f 03 0d 0b 00 52 |ion.0...|..O....R|
|00001350| 69 67 68 74 2e 20 20 54 | 68 65 20 73 6f 6c 75 74 |ight.  T|he solut|
|00001360| 69 6f 6e 20 69 73 20 31 | 2f 32 2e 0d 0a 00 00 17 |ion is 1|/2......|
|00001370| 4f 0c 0d 0b 00 12 33 57 | 72 6f 6e 67 2e 20 20 57 |O.....3W|rong.  W|
|00001380| 65 20 73 6f 6c 76 65 20 | 74 68 65 20 65 71 75 61 |e solve |the equa|
|00001390| 74 69 6f 6e 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |tion as |follows.|
|000013a0| 12 30 0d 0a 00 20 20 20 | 20 20 11 34 44 32 32 32 |.0...   |  .4D222|
|000013b0| 32 32 32 20 20 20 20 20 | 20 20 20 44 32 32 0d 0b |222     |   D22..|
|000013c0| 00 20 20 20 20 53 20 11 | 31 32 11 33 74 20 11 31 |.    S .|12.3t .1|
|000013d0| 2b 20 33 20 3d 20 33 20 | 2d 20 11 34 53 20 11 31 |+ 3 = 3 |- .4S .1|
|000013e0| 32 11 33 74 20 20 20 20 | 20 20 20 20 20 20 20 20 |2.3t    |        |
|000013f0| 20 20 20 20 20 20 0d 0a | 00 20 20 20 20 20 20 20 |      ..|.       |
|00001400| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 34 |        |      .4|
|00001410| 44 32 32 0d 0b 00 20 20 | 20 20 20 20 11 31 32 11 |D22...  |    .12.|
|00001420| 33 74 20 11 31 2b 20 33 | 20 3d 20 39 20 2d 20 36 |3t .1+ 3| = 9 - 6|
|00001430| 11 34 53 20 11 31 32 11 | 33 74 20 11 31 2b 20 32 |.4S .12.|3t .1+ 2|
|00001440| 11 33 74 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |.3t     |       .|
|00001450| 32 12 31 53 71 75 61 72 | 65 20 62 6f 74 68 20 73 |2.1Squar|e both s|
|00001460| 69 64 65 73 12 30 0d 0a | 00 20 20 20 20 20 20 20 |ides.0..|.       |
|00001470| 20 20 20 20 20 20 20 20 | 20 20 20 11 34 44 32 32 |        |   .4D22|
|00001480| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 11 31 2d |...     |     .1-|
|00001490| 36 20 3d 20 2d 36 11 34 | 53 20 11 31 32 11 33 74 |6 = -6.4|S .12.3t|
|000014a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000014b0| 20 20 20 20 11 32 12 31 | 49 73 6f 6c 61 74 65 20 |    .2.1|Isolate |
|000014c0| 72 61 64 69 63 61 6c 12 | 30 0d 0a 00 20 20 20 20 |radical.|0...    |
|000014d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 34 44 32 |        |    .4D2|
|000014e0| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 11 |2...    |       .|
|000014f0| 31 31 20 3d 20 11 34 53 | 20 11 31 32 11 33 74 20 |11 = .4S| .12.3t |
|00001500| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001510| 20 20 20 20 20 11 32 12 | 31 44 69 76 69 64 65 20 |     .2.|1Divide |
|00001520| 62 6f 74 68 20 73 69 64 | 65 73 20 62 79 20 2d 36 |both sid|es by -6|
|00001530| 12 30 0d 0a 00 0d 0b 00 | 20 20 20 20 20 20 20 20 |.0......|        |
|00001540| 20 20 20 11 31 31 20 3d | 20 32 11 33 74 20 20 20 |   .11 =| 2.3t   |
|00001550| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001560| 20 20 20 20 20 11 32 12 | 31 53 71 75 61 72 65 20 |     .2.|1Square |
|00001570| 62 6f 74 68 20 73 69 64 | 65 73 12 30 0d 0a 00 0d |both sid|es.0....|
|00001580| 0b 00 20 20 20 20 20 20 | 20 20 20 11 31 31 2f 32 |..      |   .11/2|
|00001590| 20 3d 20 11 33 74 20 20 | 20 20 20 20 20 20 20 20 | = .3t  |        |
|000015a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000015b0| 32 12 31 53 6f 6c 75 74 | 69 6f 6e 12 30 0d 0a 00 |2.1Solut|ion.0...|
|000015c0| 00 17 4f 03 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 54 |..O....W|rong.  T|
|000015d0| 68 69 73 20 65 71 75 61 | 74 69 6f 6e 20 68 61 73 |his equa|tion has|
|000015e0| 20 61 20 72 65 61 6c 20 | 73 6f 6c 75 74 69 6f 6e | a real |solution|
|000015f0| 2e 20 20 54 72 79 20 61 | 67 61 69 6e 2e 0d 0a 00 |.  Try a|gain....|
|00001600| 00 17 4f 0a 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 4c |..O....W|rong.  L|
|00001610| 65 74 74 69 6e 67 20 11 | 33 70 20 11 31 3d 20 33 |etting .|3p .1= 3|
|00001620| 34 2e 37 2c 20 77 65 20 | 73 6f 6c 76 65 20 61 73 |4.7, we |solve as|
|00001630| 20 66 6f 6c 6c 6f 77 73 | 2e 12 30 0d 0a 00 20 20 | follows|..0...  |
|00001640| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001650| 20 20 20 20 20 11 34 44 | 32 32 32 32 32 32 32 32 |     .4D|22222222|
|00001660| 32 32 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 11 |22...   |       .|
|00001670| 31 33 34 2e 37 20 3d 20 | 35 31 20 2d 20 11 34 53 |134.7 = |51 - .4S|
|00001680| 20 11 31 30 2e 30 30 31 | 11 33 78 20 11 31 2b 20 | .10.001|.3x .1+ |
|00001690| 31 20 20 20 20 20 20 11 | 32 12 31 47 69 76 65 6e |1      .|2.1Given|
|000016a0| 20 65 71 75 61 74 69 6f | 6e 20 77 69 74 68 20 70 | equatio|n with p|
|000016b0| 3d 33 34 2e 37 12 30 0d | 0a 00 20 20 20 11 34 44 |=34.7.0.|..   .4D|
|000016c0| 32 32 32 32 32 32 32 32 | 32 32 0d 0b 00 20 20 53 |22222222|22...  S|
|000016d0| 20 11 31 30 2e 30 30 31 | 11 33 78 20 11 31 2b 20 | .10.001|.3x .1+ |
|000016e0| 31 20 3d 20 31 36 2e 33 | 20 20 20 20 20 20 20 20 |1 = 16.3|        |
|000016f0| 20 20 20 20 20 20 20 20 | 20 20 20 0d 0a 00 20 20 |        |   ...  |
|00001700| 20 20 30 2e 30 30 31 11 | 33 78 20 11 31 2b 20 31 |  0.001.|3x .1+ 1|
|00001710| 20 3d 20 32 36 35 2e 36 | 39 20 20 20 20 20 20 20 | = 265.6|9       |
|00001720| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 53 71 |        |  .2.1Sq|
|00001730| 75 61 72 65 20 62 6f 74 | 68 20 73 69 64 65 73 12 |uare bot|h sides.|
|00001740| 30 0d 0a 00 20 20 20 20 | 20 20 20 20 11 31 30 2e |0...    |    .10.|
|00001750| 30 30 31 11 33 78 20 11 | 31 3d 20 32 36 34 2e 36 |001.3x .|1= 264.6|
|00001760| 39 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |9       |        |
|00001770| 20 20 11 32 12 31 53 75 | 62 74 72 61 63 74 20 31 |  .2.1Su|btract 1|
|00001780| 20 66 72 6f 6d 20 62 6f | 74 68 20 73 69 64 65 73 | from bo|th sides|
|00001790| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|000017a0| 20 20 11 33 78 20 11 31 | 3d 20 32 36 34 2c 36 39 |  .3x .1|= 264,69|
|000017b0| 30 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |0       |        |
|000017c0| 20 11 32 12 31 44 69 76 | 69 64 65 20 62 79 20 30 | .2.1Div|ide by 0|
|000017d0| 2e 30 30 31 12 30 0d 0a | 00 0d 0b 00 11 31 12 33 |.001.0..|.....1.3|
|000017e0| 54 68 75 73 2c 20 77 68 | 65 6e 20 74 68 65 20 70 |Thus, wh|en the p|
|000017f0| 72 69 63 65 20 69 73 20 | 24 33 34 2e 37 30 2c 20 |rice is |$34.70, |
|00001800| 74 68 65 20 64 65 6d 61 | 6e 64 20 70 65 72 20 64 |the dema|nd per d|
|00001810| 61 79 20 69 73 20 32 36 | 34 2c 36 39 30 2e 0d 0a |ay is 26|4,690...|
|00001820| 00 00 17 4f 03 0d 0b 00 | 52 69 67 68 74 2e 20 20 |...O....|Right.  |
|00001830| 57 68 65 6e 20 74 68 65 | 20 70 72 69 63 65 20 69 |When the| price i|
|00001840| 73 20 24 33 34 2e 37 30 | 2c 20 74 68 65 20 64 65 |s $34.70|, the de|
|00001850| 6d 61 6e 64 20 70 65 72 | 20 64 61 79 20 69 73 20 |mand per| day is |
|00001860| 32 36 34 2c 36 39 30 2e | 0d 0a 00 00 17 4f 0a 0d |264,690.|.....O..|
|00001870| 0b 00 57 72 6f 6e 67 2e | 20 20 4c 65 74 74 69 6e |..Wrong.|  Lettin|
|00001880| 67 20 11 33 70 20 11 31 | 3d 20 33 34 2e 37 2c 20 |g .3p .1|= 34.7, |
|00001890| 77 65 20 73 6f 6c 76 65 | 20 61 73 20 66 6f 6c 6c |we solve| as foll|
|000018a0| 6f 77 73 2e 12 30 0d 0a | 00 20 20 20 20 20 20 20 |ows..0..|.       |
|000018b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000018c0| 11 34 44 32 32 32 32 32 | 32 32 32 32 32 0d 0b 00 |.4D22222|22222...|
|000018d0| 20 20 20 20 20 20 20 20 | 20 20 11 31 33 34 2e 37 |        |  .134.7|
|000018e0| 20 3d 20 35 31 20 2d 20 | 11 34 53 20 11 31 30 2e | = 51 - |.4S .10.|
|000018f0| 30 30 31 11 33 78 20 11 | 31 2b 20 31 20 20 20 20 |001.3x .|1+ 1    |
|00001900| 20 20 11 32 12 31 47 69 | 76 65 6e 20 65 71 75 61 |  .2.1Gi|ven equa|
|00001910| 74 69 6f 6e 20 77 69 74 | 68 20 70 3d 33 34 2e 37 |tion wit|h p=34.7|
|00001920| 12 30 0d 0a 00 20 20 20 | 11 34 44 32 32 32 32 32 |.0...   |.4D22222|
|00001930| 32 32 32 32 32 0d 0b 00 | 20 20 53 20 11 31 30 2e |22222...|  S .10.|
|00001940| 30 30 31 11 33 78 20 11 | 31 2b 20 31 20 3d 20 31 |001.3x .|1+ 1 = 1|
|00001950| 36 2e 33 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |6.3     |        |
|00001960| 20 20 20 20 20 20 0d 0a | 00 20 20 20 20 30 2e 30 |      ..|.    0.0|
|00001970| 30 31 11 33 78 20 11 31 | 2b 20 31 20 3d 20 32 36 |01.3x .1|+ 1 = 26|
|00001980| 35 2e 36 39 20 20 20 20 | 20 20 20 20 20 20 20 20 |5.69    |        |
|00001990| 20 20 20 20 20 11 32 12 | 31 53 71 75 61 72 65 20 |     .2.|1Square |
|000019a0| 62 6f 74 68 20 73 69 64 | 65 73 12 30 0d 0a 00 20 |both sid|es.0... |
|000019b0| 20 20 20 20 20 20 20 11 | 31 30 2e 30 30 31 11 33 |       .|10.001.3|
|000019c0| 78 20 11 31 3d 20 32 36 | 34 2e 36 39 20 20 20 20 |x .1= 26|4.69    |
|000019d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|000019e0| 31 53 75 62 74 72 61 63 | 74 20 31 20 66 72 6f 6d |1Subtrac|t 1 from|
|000019f0| 20 62 6f 74 68 20 73 69 | 64 65 73 12 30 0d 0a 00 | both si|des.0...|
|00001a00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |        |     .3x|
|00001a10| 20 11 31 3d 20 32 36 34 | 2c 36 39 30 20 20 20 20 | .1= 264|,690    |
|00001a20| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00001a30| 44 69 76 69 64 65 20 62 | 79 20 30 2e 30 30 31 12 |Divide b|y 0.001.|
|00001a40| 30 0d 0a 00 0d 0b 00 11 | 31 12 33 54 68 75 73 2c |0.......|1.3Thus,|
|00001a50| 20 77 68 65 6e 20 74 68 | 65 20 70 72 69 63 65 20 | when th|e price |
|00001a60| 69 73 20 24 33 34 2e 37 | 30 2c 20 74 68 65 20 64 |is $34.7|0, the d|
|00001a70| 65 6d 61 6e 64 20 70 65 | 72 20 64 61 79 20 69 73 |emand pe|r day is|
|00001a80| 20 32 36 34 2c 36 39 30 | 2e 0d 0a 00 00 17 4f 0a | 264,690|......O.|
|00001a90| 0d 0b 00 57 72 6f 6e 67 | 2e 20 20 4c 65 74 74 69 |...Wrong|.  Letti|
|00001aa0| 6e 67 20 11 33 70 20 11 | 31 3d 20 33 34 2e 37 2c |ng .3p .|1= 34.7,|
|00001ab0| 20 77 65 20 73 6f 6c 76 | 65 20 61 73 20 66 6f 6c | we solv|e as fol|
|00001ac0| 6c 6f 77 73 2e 12 30 0d | 0a 00 20 20 20 20 20 20 |lows..0.|..      |
|00001ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ae0| 20 11 34 44 32 32 32 32 | 32 32 32 32 32 32 0d 0b | .4D2222|222222..|
|00001af0| 00 20 20 20 20 20 20 20 | 20 20 20 11 31 33 34 2e |.       |   .134.|
|00001b00| 37 20 3d 20 35 31 20 2d | 20 11 34 53 20 11 31 30 |7 = 51 -| .4S .10|
|00001b10| 2e 30 30 31 11 33 78 20 | 11 31 2b 20 31 20 20 20 |.001.3x |.1+ 1   |
|00001b20| 20 20 20 11 32 12 31 47 | 69 76 65 6e 20 65 71 75 |   .2.1G|iven equ|
|00001b30| 61 74 69 6f 6e 20 77 69 | 74 68 20 70 3d 33 34 2e |ation wi|th p=34.|
|00001b40| 37 12 30 0d 0a 00 20 20 | 20 11 34 44 32 32 32 32 |7.0...  | .4D2222|
|00001b50| 32 32 32 32 32 32 0d 0b | 00 20 20 53 20 11 31 30 |222222..|.  S .10|
|00001b60| 2e 30 30 31 11 33 78 20 | 11 31 2b 20 31 20 3d 20 |.001.3x |.1+ 1 = |
|00001b70| 31 36 2e 33 20 20 20 20 | 20 20 20 20 20 20 20 20 |16.3    |        |
|00001b80| 20 20 20 20 20 20 20 0d | 0a 00 20 20 20 20 30 2e |       .|..    0.|
|00001b90| 30 30 31 11 33 78 20 11 | 31 2b 20 31 20 3d 20 32 |001.3x .|1+ 1 = 2|
|00001ba0| 36 35 2e 36 39 20 20 20 | 20 20 20 20 20 20 20 20 |65.69   |        |
|00001bb0| 20 20 20 20 20 20 11 32 | 12 31 53 71 75 61 72 65 |      .2|.1Square|
|00001bc0| 20 62 6f 74 68 20 73 69 | 64 65 73 12 30 0d 0a 00 | both si|des.0...|
|00001bd0| 20 20 20 20 20 20 20 20 | 11 31 30 2e 30 30 31 11 |        |.10.001.|
|00001be0| 33 78 20 11 31 3d 20 32 | 36 34 2e 36 39 20 20 20 |3x .1= 2|64.69   |
|00001bf0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00001c00| 12 31 53 75 62 74 72 61 | 63 74 20 31 20 66 72 6f |.1Subtra|ct 1 fro|
|00001c10| 6d 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0a |m both s|ides.0..|
|00001c20| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |.       |      .3|
|00001c30| 78 20 11 31 3d 20 32 36 | 34 2c 36 39 30 20 20 20 |x .1= 26|4,690   |
|00001c40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00001c50| 31 44 69 76 69 64 65 20 | 62 79 20 30 2e 30 30 31 |1Divide |by 0.001|
|00001c60| 12 30 0d 0a 00 0d 0b 00 | 11 31 12 33 54 68 75 73 |.0......|.1.3Thus|
|00001c70| 2c 20 77 68 65 6e 20 74 | 68 65 20 70 72 69 63 65 |, when t|he price|
|00001c80| 20 69 73 20 24 33 34 2e | 37 30 2c 20 74 68 65 20 | is $34.|70, the |
|00001c90| 64 65 6d 61 6e 64 20 70 | 65 72 20 64 61 79 20 69 |demand p|er day i|
|00001ca0| 73 20 32 36 34 2c 36 39 | 30 2e 0d 0a 00 00 17 4f |s 264,69|0......O|
|00001cb0| 07 0d 0b 00 4e 6f 2e 20 | 20 54 68 69 73 20 65 71 |....No. | This eq|
|00001cc0| 75 61 74 69 6f 6e 20 68 | 61 73 20 6f 6e 6c 79 20 |uation h|as only |
|00001cd0| 74 77 6f 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f |two real| solutio|
|00001ce0| 6e 73 3a 20 33 20 61 6e | 64 20 2d 33 2e 20 20 54 |ns: 3 an|d -3.  T|
|00001cf0| 6f 20 73 65 65 20 77 68 | 79 2c 20 77 65 0d 0a 00 |o see wh|y, we...|
|00001d00| 20 20 20 20 20 20 20 20 | 11 32 34 20 20 20 20 20 |        |.24     |
|00001d10| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|00001d20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 32 0d |        |      2.|
|00001d30| 0b 00 11 31 66 61 63 74 | 6f 72 20 11 33 78 20 20 |...1fact|or .3x  |
|00001d40| 11 31 2d 20 37 11 33 78 | 20 20 11 31 2d 20 31 38 |.1- 7.3x|  .1- 18|
|00001d50| 20 3d 20 30 20 61 73 20 | 28 11 33 78 20 11 31 2d | = 0 as |(.3x .1-|
|00001d60| 20 33 29 28 11 33 78 20 | 11 31 2b 20 33 29 28 11 | 3)(.3x |.1+ 3)(.|
|00001d70| 33 78 20 20 11 31 2b 20 | 32 29 20 3d 20 30 2e 20 |3x  .1+ |2) = 0. |
|00001d80| 20 53 65 74 74 69 6e 67 | 20 65 61 63 68 20 0d 0a | Setting| each ..|
|00001d90| 00 0d 0b 00 66 61 63 74 | 6f 72 20 65 71 75 61 6c |....fact|or equal|
|00001da0| 20 74 6f 20 7a 65 72 6f | 20 61 6e 64 20 73 6f 6c | to zero| and sol|
|00001db0| 76 69 6e 67 20 66 6f 72 | 20 11 33 78 11 31 2c 20 |ving for| .3x.1, |
|00001dc0| 77 65 20 6f 62 74 61 69 | 6e 20 11 33 78 20 11 31 |we obtai|n .3x .1|
|00001dd0| 3d 20 33 2c 20 11 33 78 | 20 11 31 3d 20 2d 33 2c |= 3, .3x| .1= -3,|
|00001de0| 20 61 6e 64 20 74 68 65 | 20 0d 0a 00 0d 0b 00 74 | and the| ......t|
|00001df0| 68 69 72 64 20 66 61 63 | 74 6f 72 20 79 69 65 6c |hird fac|tor yiel|
|00001e00| 64 73 20 6e 6f 20 72 65 | 61 6c 20 73 6f 6c 75 74 |ds no re|al solut|
|00001e10| 69 6f 6e 2e 0d 0a 00 00 | 17 4f 03 0d 0b 00 52 69 |ion.....|.O....Ri|
|00001e20| 67 68 74 2e 20 20 54 72 | 79 20 63 68 65 63 6b 69 |ght.  Tr|y checki|
|00001e30| 6e 67 20 65 61 63 68 20 | 6f 66 20 74 68 65 73 65 |ng each |of these|
|00001e40| 20 73 6f 6c 75 74 69 6f | 6e 73 20 69 6e 20 74 68 | solutio|ns in th|
|00001e50| 65 20 6f 72 69 67 69 6e | 61 6c 20 65 71 75 61 74 |e origin|al equat|
|00001e60| 69 6f 6e 2e 0d 0a 00 00 | 17 4f 07 0d 0b 00 4e 6f |ion.....|.O....No|
|00001e70| 2e 20 20 54 68 69 73 20 | 65 71 75 61 74 69 6f 6e |.  This |equation|
|00001e80| 20 68 61 73 20 6f 6e 6c | 79 20 74 77 6f 20 72 65 | has onl|y two re|
|00001e90| 61 6c 20 73 6f 6c 75 74 | 69 6f 6e 73 3a 20 33 20 |al solut|ions: 3 |
|00001ea0| 61 6e 64 20 2d 33 2e 20 | 20 54 6f 20 73 65 65 20 |and -3. | To see |
|00001eb0| 77 68 79 2c 20 77 65 0d | 0a 00 20 20 20 20 20 20 |why, we.|..      |
|00001ec0| 20 20 11 32 34 20 20 20 | 20 20 32 20 20 20 20 20 |  .24   |  2     |
|00001ed0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ee0| 20 20 20 20 20 20 20 20 | 32 0d 0b 00 11 31 66 61 |        |2....1fa|
|00001ef0| 63 74 6f 72 20 11 33 78 | 20 20 11 31 2d 20 37 11 |ctor .3x|  .1- 7.|
|00001f00| 33 78 20 20 11 31 2d 20 | 31 38 20 3d 20 30 20 61 |3x  .1- |18 = 0 a|
|00001f10| 73 20 28 11 33 78 20 11 | 31 2d 20 33 29 28 11 33 |s (.3x .|1- 3)(.3|
|00001f20| 78 20 11 31 2b 20 33 29 | 28 11 33 78 20 20 11 31 |x .1+ 3)|(.3x  .1|
|00001f30| 2b 20 32 29 20 3d 20 30 | 2e 20 20 53 65 74 74 69 |+ 2) = 0|.  Setti|
|00001f40| 6e 67 20 65 61 63 68 20 | 0d 0a 00 0d 0b 00 66 61 |ng each |......fa|
|00001f50| 63 74 6f 72 20 65 71 75 | 61 6c 20 74 6f 20 7a 65 |ctor equ|al to ze|
|00001f60| 72 6f 20 61 6e 64 20 73 | 6f 6c 76 69 6e 67 20 66 |ro and s|olving f|
|00001f70| 6f 72 20 11 33 78 11 31 | 2c 20 77 65 20 6f 62 74 |or .3x.1|, we obt|
|00001f80| 61 69 6e 20 11 33 78 20 | 11 31 3d 20 33 2c 20 11 |ain .3x |.1= 3, .|
|00001f90| 33 78 20 11 31 3d 20 2d | 33 2c 20 61 6e 64 20 74 |3x .1= -|3, and t|
|00001fa0| 68 65 20 0d 0a 00 0d 0b | 00 74 68 69 72 64 20 66 |he .....|.third f|
|00001fb0| 61 63 74 6f 72 20 79 69 | 65 6c 64 73 20 6e 6f 20 |actor yi|elds no |
|00001fc0| 72 65 61 6c 20 73 6f 6c | 75 74 69 6f 6e 2e 0d 0a |real sol|ution...|
|00001fd0| 00 00 17 4f 03 0d 0b 00 | 57 72 6f 6e 67 2e 20 20 |...O....|Wrong.  |
|00001fe0| 54 68 65 20 67 69 76 65 | 6e 20 65 71 75 61 74 69 |The give|n equati|
|00001ff0| 6f 6e 20 68 61 73 20 74 | 77 6f 20 72 65 61 6c 20 |on has t|wo real |
|00002000| 73 6f 6c 75 74 69 6f 6e | 73 2e 20 20 54 72 79 20 |solution|s.  Try |
|00002010| 61 67 61 69 6e 2e 0d 0a | 00 00 17 4f 0a 20 20 20 |again...|...O.   |
|00002020| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002030| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002040| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002050| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 33 |        |     .23|
|00002060| 0d 0b 00 11 31 57 72 6f | 6e 67 2e 20 20 53 69 6e |....1Wro|ng.  Sin|
|00002070| 63 65 20 74 68 65 20 67 | 69 76 65 6e 20 65 71 75 |ce the g|iven equ|
|00002080| 61 74 69 6f 6e 20 69 73 | 20 6f 66 20 71 75 61 64 |ation is| of quad|
|00002090| 72 61 74 69 63 20 74 79 | 70 65 20 77 69 74 68 20 |ratic ty|pe with |
|000020a0| 11 33 75 20 11 31 3d 20 | 11 33 78 20 11 31 2c 20 |.3u .1= |.3x .1, |
|000020b0| 77 65 20 73 6f 6c 76 65 | 0d 0a 00 61 73 20 66 6f |we solve|...as fo|
|000020c0| 6c 6c 6f 77 73 2e 20 20 | 12 30 0d 0a 00 20 20 20 |llows.  |.0...   |
|000020d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000020e0| 20 20 20 20 20 20 20 11 | 32 36 20 20 20 20 20 20 |       .|26      |
|000020f0| 33 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |3...    |        |
|00002100| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |        |     .3x|
|00002110| 20 20 11 31 2b 20 32 36 | 11 33 78 20 20 11 31 2d |  .1+ 26|.3x  .1-|
|00002120| 20 32 37 20 3d 20 30 0d | 0a 00 20 20 20 20 20 20 | 27 = 0.|..      |
|00002130| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002140| 20 20 11 32 33 20 20 20 | 20 20 20 20 20 33 0d 0b |  .23   |     3..|
|00002150| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00002160| 20 20 20 20 20 20 20 11 | 31 28 11 33 78 20 20 11 |       .|1(.3x  .|
|00002170| 31 2b 20 32 37 29 28 11 | 33 78 20 20 11 31 2d 20 |1+ 27)(.|3x  .1- |
|00002180| 31 29 20 3d 20 30 0d 0a | 00 20 20 20 20 20 20 20 |1) = 0..|.       |
|00002190| 20 20 11 32 32 20 20 20 | 20 20 20 20 20 20 20 20 |  .22   |        |
|000021a0| 20 20 20 20 20 20 20 20 | 32 0d 0b 00 11 31 28 11 |        |2....1(.|
|000021b0| 33 78 20 11 31 2b 20 33 | 29 28 11 33 78 20 20 11 |3x .1+ 3|)(.3x  .|
|000021c0| 31 2d 20 33 11 33 78 20 | 11 31 2b 20 39 29 28 11 |1- 3.3x |.1+ 9)(.|
|000021d0| 33 78 20 11 31 2d 20 31 | 29 28 11 33 78 20 20 11 |3x .1- 1|)(.3x  .|
|000021e0| 31 2b 20 11 33 78 20 11 | 31 2b 20 31 29 20 3d 20 |1+ .3x .|1+ 1) = |
|000021f0| 30 0d 0a 00 12 33 53 65 | 74 74 69 6e 67 20 65 61 |0....3Se|tting ea|
|00002200| 63 68 20 66 61 63 74 6f | 72 20 65 71 75 61 6c 20 |ch facto|r equal |
|00002210| 74 6f 20 7a 65 72 6f 2c | 20 74 68 65 20 73 6f 6c |to zero,| the sol|
|00002220| 75 74 69 6f 6e 73 20 61 | 72 65 3a 20 11 33 78 20 |utions a|re: .3x |
|00002230| 11 31 3d 20 2d 33 2c 20 | 6e 6f 20 72 65 61 6c 0d |.1= -3, |no real.|
|00002240| 0a 00 73 6f 6c 75 74 69 | 6f 6e 2c 20 11 33 78 20 |..soluti|on, .3x |
|00002250| 11 31 3d 20 31 2c 20 61 | 6e 64 20 6e 6f 20 72 65 |.1= 1, a|nd no re|
|00002260| 61 6c 20 73 6f 6c 75 74 | 69 6f 6e 2e 0d 0a 00 00 |al solut|ion.....|
|00002270| 17 4f 0a 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.O.     |        |
|00002280| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002290| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022b0| 20 20 20 11 32 33 0d 0b | 00 11 31 57 72 6f 6e 67 |   .23..|..1Wrong|
|000022c0| 2e 20 20 53 69 6e 63 65 | 20 74 68 65 20 67 69 76 |.  Since| the giv|
|000022d0| 65 6e 20 65 71 75 61 74 | 69 6f 6e 20 69 73 20 6f |en equat|ion is o|
|000022e0| 66 20 71 75 61 64 72 61 | 74 69 63 20 74 79 70 65 |f quadra|tic type|
|000022f0| 20 77 69 74 68 20 11 33 | 75 20 11 31 3d 20 11 33 | with .3|u .1= .3|
|00002300| 78 20 11 31 2c 20 77 65 | 20 73 6f 6c 76 65 0d 0a |x .1, we| solve..|
|00002310| 00 61 73 20 66 6f 6c 6c | 6f 77 73 2e 20 20 12 30 |.as foll|ows.  .0|
|00002320| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002330| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 36 |        |     .26|
|00002340| 20 20 20 20 20 20 33 0d | 0b 00 20 20 20 20 20 20 |      3.|..      |
|00002350| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002360| 20 20 20 11 33 78 20 20 | 11 31 2b 20 32 36 11 33 |   .3x  |.1+ 26.3|
|00002370| 78 20 20 11 31 2d 20 32 | 37 20 3d 20 30 0d 0a 00 |x  .1- 2|7 = 0...|
|00002380| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002390| 20 20 20 20 20 20 20 20 | 11 32 33 20 20 20 20 20 |        |.23     |
|000023a0| 20 20 20 33 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |   3... |        |
|000023b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 28 |        |     .1(|
|000023c0| 11 33 78 20 20 11 31 2b | 20 32 37 29 28 11 33 78 |.3x  .1+| 27)(.3x|
|000023d0| 20 20 11 31 2d 20 31 29 | 20 3d 20 30 0d 0a 00 20 |  .1- 1)| = 0... |
|000023e0| 20 20 20 20 20 20 20 20 | 11 32 32 20 20 20 20 20 |        |.22     |
|000023f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 32 0d |        |      2.|
|00002400| 0b 00 11 31 28 11 33 78 | 20 11 31 2b 20 33 29 28 |...1(.3x| .1+ 3)(|
|00002410| 11 33 78 20 20 11 31 2d | 20 33 11 33 78 20 11 31 |.3x  .1-| 3.3x .1|
|00002420| 2b 20 39 29 28 11 33 78 | 20 11 31 2d 20 31 29 28 |+ 9)(.3x| .1- 1)(|
|00002430| 11 33 78 20 20 11 31 2b | 20 11 33 78 20 11 31 2b |.3x  .1+| .3x .1+|
|00002440| 20 31 29 20 3d 20 30 0d | 0a 00 12 33 53 65 74 74 | 1) = 0.|...3Sett|
|00002450| 69 6e 67 20 65 61 63 68 | 20 66 61 63 74 6f 72 20 |ing each| factor |
|00002460| 65 71 75 61 6c 20 74 6f | 20 7a 65 72 6f 2c 20 74 |equal to| zero, t|
|00002470| 68 65 20 73 6f 6c 75 74 | 69 6f 6e 73 20 61 72 65 |he solut|ions are|
|00002480| 3a 20 11 33 78 20 11 31 | 3d 20 2d 33 2c 20 6e 6f |: .3x .1|= -3, no|
|00002490| 20 72 65 61 6c 0d 0a 00 | 73 6f 6c 75 74 69 6f 6e | real...|solution|
|000024a0| 2c 20 11 33 78 20 11 31 | 3d 20 31 2c 20 61 6e 64 |, .3x .1|= 1, and|
|000024b0| 20 6e 6f 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f | no real| solutio|
|000024c0| 6e 2e 0d 0a 00 00 17 4f | 03 0d 0b 00 52 69 67 68 |n......O|....Righ|
|000024d0| 74 2e 20 20 54 72 79 20 | 63 68 65 63 6b 69 6e 67 |t.  Try |checking|
|000024e0| 20 74 68 65 73 65 20 73 | 6f 6c 75 74 69 6f 6e 73 | these s|olutions|
|000024f0| 20 69 6e 20 74 68 65 20 | 6f 72 69 67 69 6e 61 6c | in the |original|
|00002500| 20 65 71 75 61 74 69 6f | 6e 2e 0d 0a 00 00 17 4f | equatio|n......O|
|00002510| 03 0d 0b 00 57 72 6f 6e | 67 2e 20 20 54 68 65 20 |....Wron|g.  The |
|00002520| 67 69 76 65 6e 20 65 71 | 75 61 74 69 6f 6e 20 68 |given eq|uation h|
|00002530| 61 73 20 74 77 6f 20 72 | 65 61 6c 20 73 6f 6c 75 |as two r|eal solu|
|00002540| 74 69 6f 6e 73 2e 20 20 | 54 72 79 20 61 67 61 69 |tions.  |Try agai|
|00002550| 6e 2e 0d 0a 00 00 17 4f | 08 0d 0b 00 57 72 6f 6e |n......O|....Wron|
|00002560| 67 2e 20 20 57 65 20 73 | 6f 6c 76 65 20 61 73 20 |g.  We s|olve as |
|00002570| 66 6f 6c 6c 6f 77 73 2e | 20 20 12 30 0d 0a 00 20 |follows.|  .0... |
|00002580| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 33 2f 32 |        |   .23/2|
|00002590| 0d 0b 00 20 20 20 20 20 | 11 31 28 11 33 78 20 11 |...     |.1(.3x .|
|000025a0| 31 2b 20 33 29 20 20 20 | 20 3d 20 38 20 20 20 20 |1+ 3)   | = 8    |
|000025b0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 47 69 76 |        | .2.1Giv|
|000025c0| 65 6e 20 65 71 75 61 74 | 69 6f 6e 12 30 0d 0a 00 |en equat|ion.0...|
|000025d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000025e0| 20 20 20 32 2f 33 0d 0b | 00 20 20 20 20 20 20 20 |   2/3..|.       |
|000025f0| 20 20 20 11 33 78 20 11 | 31 2b 20 33 20 3d 20 38 |   .3x .|1+ 3 = 8|
|00002600| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00002610| 31 52 61 69 73 65 20 62 | 6f 74 68 20 73 69 64 65 |1Raise b|oth side|
|00002620| 73 20 74 6f 20 74 68 65 | 20 32 2f 33 20 70 6f 77 |s to the| 2/3 pow|
|00002630| 65 72 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |er.0... |        |
|00002640| 20 11 33 78 20 11 31 2b | 20 33 20 3d 20 34 20 20 | .3x .1+| 3 = 4  |
|00002650| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |        |   .2.1S|
|00002660| 69 6d 70 6c 69 66 79 12 | 30 0d 0a 00 20 20 20 20 |implify.|0...    |
|00002670| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 31 |        |  .3x .1|
|00002680| 3d 20 31 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |= 1     |        |
|00002690| 11 32 12 31 53 75 62 74 | 72 61 63 74 20 33 20 66 |.2.1Subt|ract 3 f|
|000026a0| 72 6f 6d 20 62 6f 74 68 | 20 73 69 64 65 73 12 30 |rom both| sides.0|
|000026b0| 0d 0a 00 00 17 4f 03 0d | 0b 00 52 69 67 68 74 2e |.....O..|..Right.|
|000026c0| 20 20 54 72 79 20 63 68 | 65 63 6b 69 6e 67 20 74 |  Try ch|ecking t|
|000026d0| 68 69 73 20 73 6f 6c 75 | 74 69 6f 6e 20 69 6e 20 |his solu|tion in |
|000026e0| 74 68 65 20 6f 72 69 67 | 69 6e 61 6c 20 65 71 75 |the orig|inal equ|
|000026f0| 61 74 69 6f 6e 2e 0d 0a | 00 00 17 4f 08 0d 0b 00 |ation...|...O....|
|00002700| 57 72 6f 6e 67 2e 20 20 | 57 65 20 73 6f 6c 76 65 |Wrong.  |We solve|
|00002710| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 20 20 12 30 | as foll|ows.  .0|
|00002720| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |...     |       .|
|00002730| 32 33 2f 32 0d 0b 00 20 | 20 20 20 20 11 31 28 11 |23/2... |    .1(.|
|00002740| 33 78 20 11 31 2b 20 33 | 29 20 20 20 20 3d 20 38 |3x .1+ 3|)    = 8|
|00002750| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00002760| 31 47 69 76 65 6e 20 65 | 71 75 61 74 69 6f 6e 12 |1Given e|quation.|
|00002770| 30 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |0...    |        |
|00002780| 20 20 20 20 20 20 20 32 | 2f 33 0d 0b 00 20 20 20 |       2|/3...   |
|00002790| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 2b 20 33 |       .|3x .1+ 3|
|000027a0| 20 3d 20 38 20 20 20 20 | 20 20 20 20 20 20 20 20 | = 8    |        |
|000027b0| 20 11 32 12 31 52 61 69 | 73 65 20 62 6f 74 68 20 | .2.1Rai|se both |
|000027c0| 73 69 64 65 73 20 74 6f | 20 74 68 65 20 32 2f 33 |sides to| the 2/3|
|000027d0| 20 70 6f 77 65 72 12 30 | 0d 0a 00 20 20 20 20 20 | power.0|...     |
|000027e0| 20 20 20 20 20 11 33 78 | 20 11 31 2b 20 33 20 3d |     .3x| .1+ 3 =|
|000027f0| 20 34 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 | 4      |       .|
|00002800| 32 12 31 53 69 6d 70 6c | 69 66 79 12 30 0d 0a 00 |2.1Simpl|ify.0...|
|00002810| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |        |      .3|
|00002820| 78 20 11 31 3d 20 31 20 | 20 20 20 20 20 20 20 20 |x .1= 1 |        |
|00002830| 20 20 20 20 11 32 12 31 | 53 75 62 74 72 61 63 74 |    .2.1|Subtract|
|00002840| 20 33 20 66 72 6f 6d 20 | 62 6f 74 68 20 73 69 64 | 3 from |both sid|
|00002850| 65 73 12 30 0d 0a 00 00 | 17 4f 03 0d 0b 00 57 72 |es.0....|.O....Wr|
|00002860| 6f 6e 67 2e 20 20 54 68 | 65 20 67 69 76 65 6e 20 |ong.  Th|e given |
|00002870| 65 71 75 61 74 69 6f 6e | 20 68 61 73 20 6f 6e 65 |equation| has one|
|00002880| 20 72 65 61 6c 20 73 6f | 6c 75 74 69 6f 6e 2e 20 | real so|lution. |
|00002890| 20 54 72 79 20 61 67 61 | 69 6e 2e 0d 0a 00 00 17 | Try aga|in......|
|000028a0| 4f 0a 0d 0b 00 57 72 6f | 6e 67 2e 20 20 57 65 20 |O....Wro|ng.  We |
|000028b0| 73 6f 6c 76 65 20 61 73 | 20 66 6f 6c 6c 6f 77 73 |solve as| follows|
|000028c0| 2e 20 20 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |.  .0...|        |
|000028d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000028e0| 20 20 20 32 20 20 20 20 | 20 20 20 20 20 33 20 20 |   2    |     3  |
|000028f0| 20 20 20 20 20 20 31 0d | 0b 00 20 20 20 20 20 20 |      1.|..      |
|00002900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002910| 20 20 11 34 32 32 32 32 | 32 32 20 11 31 2d 20 11 |  .42222|22 .1- .|
|00002920| 34 32 32 32 32 32 32 32 | 32 32 20 11 31 3d 20 11 |42222222|22 .1= .|
|00002930| 34 32 32 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |422     |        |
|00002940| 20 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 | ...    |        |
|00002950| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 36 11 |        |    .16.|
|00002960| 33 78 20 11 31 2b 20 35 | 20 20 20 34 28 36 11 33 |3x .1+ 5|   4(6.3|
|00002970| 78 20 11 31 2b 20 35 29 | 20 20 20 32 38 0d 0a 00 |x .1+ 5)|   28...|
|00002980| 20 20 20 20 20 20 20 20 | 20 20 11 34 28 20 20 20 |        |  .4(   |
|00002990| 11 31 32 20 20 11 34 29 | 20 20 20 20 20 20 20 20 |.12  .4)|        |
|000029a0| 20 20 20 20 20 28 20 20 | 20 20 11 31 33 20 20 20 |     (  |  .13   |
|000029b0| 20 11 34 29 20 20 20 20 | 20 20 20 20 20 20 20 20 | .4)    |        |
|000029c0| 20 28 20 11 31 31 11 34 | 29 0d 0b 00 11 31 32 38 | ( .11.4|)....128|
|000029d0| 28 36 11 33 78 20 11 31 | 2b 20 35 29 11 34 21 32 |(6.3x .1|+ 5).4!2|
|000029e0| 32 32 32 32 32 21 20 11 | 31 2d 20 32 38 28 36 11 |22222! .|1- 28(6.|
|000029f0| 33 78 20 11 31 2b 20 35 | 29 11 34 21 32 32 32 32 |3x .1+ 5|).4!2222|
|00002a00| 32 32 32 32 32 21 20 11 | 31 3d 20 32 38 28 36 11 |22222! .|1= 28(6.|
|00002a10| 33 78 20 11 31 2b 20 35 | 29 11 34 21 32 32 21 20 |3x .1+ 5|).4!22! |
|00002a20| 20 20 20 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |   ...  |        |
|00002a30| 39 11 31 36 11 33 78 20 | 11 31 2b 20 35 11 34 30 |9.16.3x |.1+ 5.40|
|00002a40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 39 11 31 |        |     9.1|
|00002a50| 34 28 36 11 33 78 20 11 | 31 2b 20 35 29 11 34 30 |4(6.3x .|1+ 5).40|
|00002a60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 39 11 31 |        |     9.1|
|00002a70| 32 38 11 34 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |28.40...|        |
|00002a80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002a90| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 35 36 20 |        |   .156 |
|00002aa0| 2d 20 32 31 20 3d 20 36 | 11 33 78 20 11 31 2b 20 |- 21 = 6|.3x .1+ |
|00002ab0| 35 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |5...    |        |
|00002ac0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 33 30 20 3d |        |    30 =|
|00002ae0| 20 36 11 33 78 0d 0a 00 | 20 20 20 20 20 20 20 20 | 6.3x...|        |
|00002af0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002b00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002b10| 20 11 31 35 20 3d 20 11 | 33 78 0d 0a 00 00 17 4f | .15 = .|3x.....O|
|00002b20| 03 0d 0b 00 47 6f 6f 64 | 2e 20 20 54 72 79 20 63 |....Good|.  Try c|
|00002b30| 68 65 63 6b 69 6e 67 20 | 74 68 69 73 20 69 6e 20 |hecking |this in |
|00002b40| 74 68 65 20 6f 72 69 67 | 69 6e 61 6c 20 65 71 75 |the orig|inal equ|
|00002b50| 61 74 69 6f 6e 2e 0d 0a | 00 00 17 4f 0a 0d 0b 00 |ation...|...O....|
|00002b60| 57 72 6f 6e 67 2e 20 20 | 57 65 20 73 6f 6c 76 65 |Wrong.  |We solve|
|00002b70| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 20 20 12 30 | as foll|ows.  .0|
|00002b80| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002b90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 32 20 |        |      2 |
|00002ba0| 20 20 20 20 20 20 20 20 | 33 20 20 20 20 20 20 20 |        |3       |
|00002bb0| 20 31 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 | 1...   |        |
|00002bc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 34 32 |        |     .42|
|00002bd0| 32 32 32 32 32 20 11 31 | 2d 20 11 34 32 32 32 32 |22222 .1|- .42222|
|00002be0| 32 32 32 32 32 20 11 31 | 3d 20 11 34 32 32 20 20 |22222 .1|= .422  |
|00002bf0| 20 20 20 20 20 20 20 20 | 20 20 20 20 0d 0b 00 20 |        |    ... |
|00002c00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002c10| 20 20 20 20 20 20 20 11 | 31 36 11 33 78 20 11 31 |       .|16.3x .1|
|00002c20| 2b 20 35 20 20 20 34 28 | 36 11 33 78 20 11 31 2b |+ 5   4(|6.3x .1+|
|00002c30| 20 35 29 20 20 20 32 38 | 0d 0a 00 20 20 20 20 20 | 5)   28|...     |
|00002c40| 20 20 20 20 20 11 34 28 | 20 20 20 11 31 32 20 20 |     .4(|   .12  |
|00002c50| 11 34 29 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.4)     |        |
|00002c60| 28 20 20 20 20 11 31 33 | 20 20 20 20 11 34 29 20 |(    .13|    .4) |
|00002c70| 20 20 20 20 20 20 20 20 | 20 20 20 20 28 20 11 31 |        |    ( .1|
|00002c80| 31 11 34 29 0d 0b 00 11 | 31 32 38 28 36 11 33 78 |1.4)....|128(6.3x|
|00002c90| 20 11 31 2b 20 35 29 11 | 34 21 32 32 32 32 32 32 | .1+ 5).|4!222222|
|00002ca0| 21 20 11 31 2d 20 32 38 | 28 36 11 33 78 20 11 31 |! .1- 28|(6.3x .1|
|00002cb0| 2b 20 35 29 11 34 21 32 | 32 32 32 32 32 32 32 32 |+ 5).4!2|22222222|
|00002cc0| 21 20 11 31 3d 20 32 38 | 28 36 11 33 78 20 11 31 |! .1= 28|(6.3x .1|
|00002cd0| 2b 20 35 29 11 34 21 32 | 32 21 20 20 20 20 0d 0b |+ 5).4!2|2!    ..|
|00002ce0| 00 20 20 20 20 20 20 20 | 20 20 20 39 11 31 36 11 |.       |   9.16.|
|00002cf0| 33 78 20 11 31 2b 20 35 | 11 34 30 20 20 20 20 20 |3x .1+ 5|.40     |
|00002d00| 20 20 20 20 20 20 20 20 | 39 11 31 34 28 36 11 33 |        |9.14(6.3|
|00002d10| 78 20 11 31 2b 20 35 29 | 11 34 30 20 20 20 20 20 |x .1+ 5)|.40     |
|00002d20| 20 20 20 20 20 20 20 20 | 39 11 31 32 38 11 34 30 |        |9.128.40|
|00002d30| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002d40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d50| 20 20 20 20 20 20 11 31 | 35 36 20 2d 20 32 31 20 |      .1|56 - 21 |
|00002d60| 3d 20 36 11 33 78 20 11 | 31 2b 20 35 0d 0a 00 20 |= 6.3x .|1+ 5... |
|00002d70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002d90| 20 20 20 20 20 20 20 33 | 30 20 3d 20 36 11 33 78 |       3|0 = 6.3x|
|00002da0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00002db0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002dc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 35 20 |        |    .15 |
|00002dd0| 3d 20 11 33 78 0d 0a 00 | 00 17 4f 03 0d 0b 00 57 |= .3x...|..O....W|
|00002de0| 72 6f 6e 67 2e 20 20 54 | 68 65 20 67 69 76 65 6e |rong.  T|he given|
|00002df0| 20 65 71 75 61 74 69 6f | 6e 20 68 61 73 20 6f 6e | equatio|n has on|
|00002e00| 65 20 72 65 61 6c 20 73 | 6f 6c 75 74 69 6f 6e 2e |e real s|olution.|
|00002e10| 20 20 54 72 79 20 61 67 | 61 69 6e 2e 0d 0a 00 00 |  Try ag|ain.....|
|00002e20| 17 4f 03 0d 0b 00 52 69 | 67 68 74 2e 20 20 54 68 |.O....Ri|ght.  Th|
|00002e30| 65 20 67 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |e given |equation|
|00002e40| 20 68 61 73 20 74 77 6f | 20 72 65 61 6c 20 73 6f | has two| real so|
|00002e50| 6c 75 74 69 6f 6e 73 3a | 20 32 20 61 6e 64 20 2d |lutions:| 2 and -|
|00002e60| 32 2e 0d 0a 00 00 17 4f | 09 0d 0b 00 57 72 6f 6e |2......O|....Wron|
|00002e70| 67 2e 20 20 57 65 20 73 | 6f 6c 76 65 20 61 73 20 |g.  We s|olve as |
|00002e80| 66 6f 6c 6c 6f 77 73 2e | 20 12 30 0d 0a 00 20 20 |follows.| .0...  |
|00002e90| 20 20 20 20 20 20 20 20 | 11 32 34 0d 0b 00 20 20 |        |.24...  |
|00002ea0| 20 20 20 20 20 20 20 11 | 33 78 20 20 11 31 2d 20 |       .|3x  .1- |
|00002eb0| 31 36 20 3d 20 30 20 20 | 20 20 20 20 20 20 20 20 |16 = 0  |        |
|00002ec0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ed0| 20 20 20 20 11 32 12 31 | 47 69 76 65 6e 20 65 71 |    .2.1|Given eq|
|00002ee0| 75 61 74 69 6f 6e 12 30 | 0d 0a 00 20 20 32 20 20 |uation.0|...  2  |
|00002ef0| 20 20 20 20 20 32 0d 0b | 00 11 31 28 11 33 78 20 |     2..|..1(.3x |
|00002f00| 20 11 31 2d 20 34 29 28 | 11 33 78 20 20 11 31 2b | .1- 4)(|.3x  .1+|
|00002f10| 20 34 29 20 3d 20 30 20 | 20 20 20 20 20 20 20 20 | 4) = 0 |        |
|00002f20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002f30| 20 20 20 20 20 11 32 12 | 31 46 61 63 74 6f 72 12 |     .2.|1Factor.|
|00002f40| 30 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 32 |0...    |       2|
|00002f50| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |...     |     .3x|
|00002f60| 20 20 11 31 2d 20 34 20 | 3d 20 30 20 20 20 11 34 |  .1- 4 |= 0   .4|
|00002f70| 35 35 36 20 20 20 11 33 | 78 20 11 31 3d 20 11 34 |556   .3|x .1= .4|
|00002f80| 2b 11 31 32 20 20 20 20 | 20 20 20 20 20 20 20 20 |+.12    |        |
|00002f90| 20 20 20 11 32 12 31 53 | 65 74 20 31 73 74 20 66 |   .2.1S|et 1st f|
|00002fa0| 61 63 74 6f 72 20 65 71 | 75 61 6c 20 74 6f 20 30 |actor eq|ual to 0|
|00002fb0| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|00002fc0| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 11 33 |2...    |      .3|
|00002fd0| 78 20 20 11 31 2b 20 34 | 20 3d 20 30 20 20 20 11 |x  .1+ 4| = 0   .|
|00002fe0| 34 35 35 36 20 20 20 11 | 31 4e 6f 20 72 65 61 6c |4556   .|1No real|
|00002ff0| 20 73 6f 6c 75 74 69 6f | 6e 20 20 20 20 20 11 32 | solutio|n     .2|
|00003000| 12 31 53 65 74 20 32 6e | 64 20 66 61 63 74 6f 72 |.1Set 2n|d factor|
|00003010| 20 65 71 75 61 6c 20 74 | 6f 20 30 12 30 0d 0a 00 | equal t|o 0.0...|
|00003020| 00 17 4f 09 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 57 |..O....W|rong.  W|
|00003030| 65 20 73 6f 6c 76 65 20 | 61 73 20 66 6f 6c 6c 6f |e solve |as follo|
|00003040| 77 73 2e 20 12 30 0d 0a | 00 20 20 20 20 20 20 20 |ws. .0..|.       |
|00003050| 20 20 20 11 32 34 0d 0b | 00 20 20 20 20 20 20 20 |   .24..|.       |
|00003060| 20 20 11 33 78 20 20 11 | 31 2d 20 31 36 20 3d 20 |  .3x  .|1- 16 = |
|00003070| 30 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |0       |        |
|00003080| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00003090| 32 12 31 47 69 76 65 6e | 20 65 71 75 61 74 69 6f |2.1Given| equatio|
|000030a0| 6e 12 30 0d 0a 00 20 20 | 32 20 20 20 20 20 20 20 |n.0...  |2       |
|000030b0| 32 0d 0b 00 11 31 28 11 | 33 78 20 20 11 31 2d 20 |2....1(.|3x  .1- |
|000030c0| 34 29 28 11 33 78 20 20 | 11 31 2b 20 34 29 20 3d |4)(.3x  |.1+ 4) =|
|000030d0| 20 30 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | 0      |        |
|000030e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000030f0| 11 32 12 31 46 61 63 74 | 6f 72 12 30 0d 0a 00 20 |.2.1Fact|or.0... |
|00003100| 20 20 20 20 20 20 20 20 | 20 20 32 0d 0b 00 20 20 |        |  2...  |
|00003110| 20 20 20 20 20 20 20 20 | 11 33 78 20 20 11 31 2d |        |.3x  .1-|
|00003120| 20 34 20 3d 20 30 20 20 | 20 11 34 35 35 36 20 20 | 4 = 0  | .4556  |
|00003130| 20 11 33 78 20 11 31 3d | 20 11 34 2b 11 31 32 20 | .3x .1=| .4+.12 |
|00003140| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00003150| 12 31 53 65 74 20 31 73 | 74 20 66 61 63 74 6f 72 |.1Set 1s|t factor|
|00003160| 20 65 71 75 61 6c 20 74 | 6f 20 30 12 30 0d 0a 00 | equal t|o 0.0...|
|00003170| 20 20 20 20 20 20 20 20 | 20 20 20 32 0d 0b 00 20 |        |   2... |
|00003180| 20 20 20 20 20 20 20 20 | 20 11 33 78 20 20 11 31 |        | .3x  .1|
|00003190| 2b 20 34 20 3d 20 30 20 | 20 20 11 34 35 35 36 20 |+ 4 = 0 |  .4556 |
|000031a0| 20 20 11 31 4e 6f 20 72 | 65 61 6c 20 73 6f 6c 75 |  .1No r|eal solu|
|000031b0| 74 69 6f 6e 20 20 20 20 | 20 11 32 12 31 53 65 74 |tion    | .2.1Set|
|000031c0| 20 32 6e 64 20 66 61 63 | 74 6f 72 20 65 71 75 61 | 2nd fac|tor equa|
|000031d0| 6c 20 74 6f 20 30 12 30 | 0d 0a 00 00 17 4f 03 0d |l to 0.0|.....O..|
|000031e0| 0b 00 57 72 6f 6e 67 2e | 20 20 54 68 65 20 67 69 |..Wrong.|  The gi|
|000031f0| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 20 68 61 73 |ven equa|tion has|
|00003200| 20 74 77 6f 20 72 65 61 | 6c 20 73 6f 6c 75 74 69 | two rea|l soluti|
|00003210| 6f 6e 73 2e 20 20 54 72 | 79 20 61 67 61 69 6e 2e |ons.  Tr|y again.|
|00003220| 0d 0a 00 00 17 4f 03 0d | 0b 00 43 6f 72 72 65 63 |.....O..|..Correc|
|00003230| 74 2e 20 20 54 72 79 20 | 63 68 65 63 6b 69 6e 67 |t.  Try |checking|
|00003240| 20 74 68 69 73 20 73 6f | 6c 75 74 69 6f 6e 20 69 | this so|lution i|
|00003250| 6e 20 74 68 65 20 6f 72 | 69 67 69 6e 61 6c 20 65 |n the or|iginal e|
|00003260| 71 75 61 74 69 6f 6e 2e | 0d 0a 00 00 17 4f 0b 0d |quation.|.....O..|
|00003270| 0b 00 57 72 6f 6e 67 2e | 20 20 4d 75 6c 74 69 70 |..Wrong.|  Multip|
|00003280| 6c 79 20 62 79 20 74 68 | 65 20 4c 43 44 20 28 11 |ly by th|e LCD (.|
|00003290| 33 78 20 11 31 2b 20 33 | 29 28 11 33 78 20 11 31 |3x .1+ 3|)(.3x .1|
|000032a0| 2b 20 32 29 20 61 6e 64 | 20 73 6f 6c 76 65 20 61 |+ 2) and| solve a|
|000032b0| 73 20 66 6f 6c 6c 6f 77 | 73 2e 20 12 30 0d 0a 00 |s follow|s. .0...|
|000032c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 34 |        |      .4|
|000032d0| 28 20 20 20 20 20 29 20 | 20 20 20 20 20 20 20 20 |(     ) |        |
|000032e0| 20 20 20 20 20 20 20 20 | 28 20 20 20 20 20 29 0d |        |(     ).|
|000032f0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00003300| 21 20 20 11 31 34 20 20 | 11 34 21 20 20 20 20 20 |!  .14  |.4!     |
|00003310| 20 20 20 20 20 20 20 20 | 20 20 20 20 21 20 20 11 |        |    !  .|
|00003320| 31 31 20 20 11 34 21 0d | 0b 00 11 31 28 11 33 78 |11  .4!.|...1(.3x|
|00003330| 20 11 31 2b 20 33 29 28 | 11 33 78 20 11 31 2b 20 | .1+ 3)(|.3x .1+ |
|00003340| 32 29 11 34 21 32 32 32 | 32 32 21 20 11 31 2d 20 |2).4!222|22! .1- |
|00003350| 28 11 33 78 20 11 31 2b | 20 33 29 28 11 33 78 20 |(.3x .1+| 3)(.3x |
|00003360| 11 31 2b 20 32 29 11 34 | 21 32 32 32 32 32 21 20 |.1+ 2).4|!22222! |
|00003370| 11 31 3d 20 28 11 33 78 | 20 11 31 2b 20 33 29 28 |.1= (.3x| .1+ 3)(|
|00003380| 11 33 78 20 11 31 2b 20 | 32 29 31 0d 0b 00 20 20 |.3x .1+ |2)1...  |
|00003390| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 34 21 11 |        |    .4!.|
|000033a0| 33 78 20 11 31 2b 20 33 | 11 34 21 20 20 20 20 20 |3x .1+ 3|.4!     |
|000033b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 21 11 33 78 |        |    !.3x|
|000033c0| 20 11 31 2b 20 32 11 34 | 21 0d 0b 00 20 20 20 20 | .1+ 2.4|!...    |
|000033d0| 20 20 20 20 20 20 20 20 | 20 20 39 20 20 20 20 20 |        |  9     |
|000033e0| 30 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |0       |        |
|000033f0| 20 20 39 20 20 20 20 20 | 30 0d 0b 00 20 20 20 20 |  9     |0...    |
|00003400| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003410| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003420| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 32 |        |     .22|
|00003430| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00003440| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003450| 20 20 11 31 34 11 33 78 | 20 11 31 2b 20 38 20 2d |  .14.3x| .1+ 8 -|
|00003460| 20 11 33 78 20 11 31 2d | 20 33 20 3d 20 11 33 78 | .3x .1-| 3 = .3x|
|00003470| 20 20 11 31 2b 20 35 11 | 33 78 20 11 31 2b 20 36 |  .1+ 5.|3x .1+ 6|
|00003480| 2c 20 11 33 78 20 11 34 | 3d 20 11 31 2d 32 2c 20 |, .3x .4|= .1-2, |
|00003490| 2d 33 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |-3...   |        |
|000034a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000034b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000034c0| 20 20 20 20 20 20 11 32 | 32 0d 0b 00 20 20 20 20 |      .2|2...    |
|000034d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000034e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000034f0| 20 20 20 20 20 20 20 20 | 11 31 30 20 3d 20 11 33 |        |.10 = .3|
|00003500| 78 20 20 11 31 2b 20 32 | 11 33 78 20 11 31 2b 20 |x  .1+ 2|.3x .1+ |
|00003510| 31 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |1...    |        |
|00003520| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003530| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003540| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|00003550| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00003560| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003570| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 30 |        |     .10|
|00003580| 20 3d 20 28 11 33 78 20 | 11 31 2b 20 31 29 0d 0a | = (.3x |.1+ 1)..|
|00003590| 00 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |....    |        |
|000035a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000035b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 2d |        |       -|
|000035c0| 31 20 3d 20 11 33 78 0d | 0a 00 00 17 4f 0b 0d 0b |1 = .3x.|....O...|
|000035d0| 00 57 72 6f 6e 67 2e 20 | 20 4d 75 6c 74 69 70 6c |.Wrong. | Multipl|
|000035e0| 79 20 62 79 20 74 68 65 | 20 4c 43 44 20 28 11 33 |y by the| LCD (.3|
|000035f0| 78 20 11 31 2b 20 33 29 | 28 11 33 78 20 11 31 2b |x .1+ 3)|(.3x .1+|
|00003600| 20 32 29 20 61 6e 64 20 | 73 6f 6c 76 65 20 61 73 | 2) and |solve as|
|00003610| 20 66 6f 6c 6c 6f 77 73 | 2e 20 12 30 0d 0a 00 20 | follows|. .0... |
|00003620| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 34 28 |        |     .4(|
|00003630| 20 20 20 20 20 29 20 20 | 20 20 20 20 20 20 20 20 |     )  |        |
|00003640| 20 20 20 20 20 20 20 28 | 20 20 20 20 20 29 0d 0b |       (|     )..|
|00003650| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 21 |.       |       !|
|00003660| 20 20 11 31 34 20 20 11 | 34 21 20 20 20 20 20 20 |  .14  .|4!      |
|00003670| 20 20 20 20 20 20 20 20 | 20 20 20 21 20 20 11 31 |        |   !  .1|
|00003680| 31 20 20 11 34 21 0d 0b | 00 11 31 28 11 33 78 20 |1  .4!..|..1(.3x |
|00003690| 11 31 2b 20 33 29 28 11 | 33 78 20 11 31 2b 20 32 |.1+ 3)(.|3x .1+ 2|
|000036a0| 29 11 34 21 32 32 32 32 | 32 21 20 11 31 2d 20 28 |).4!2222|2! .1- (|
|000036b0| 11 33 78 20 11 31 2b 20 | 33 29 28 11 33 78 20 11 |.3x .1+ |3)(.3x .|
|000036c0| 31 2b 20 32 29 11 34 21 | 32 32 32 32 32 21 20 11 |1+ 2).4!|22222! .|
|000036d0| 31 3d 20 28 11 33 78 20 | 11 31 2b 20 33 29 28 11 |1= (.3x |.1+ 3)(.|
|000036e0| 33 78 20 11 31 2b 20 32 | 29 31 0d 0b 00 20 20 20 |3x .1+ 2|)1...   |
|000036f0| 20 20 20 20 20 20 20 20 | 20 20 20 11 34 21 11 33 |        |   .4!.3|
|00003700| 78 20 11 31 2b 20 33 11 | 34 21 20 20 20 20 20 20 |x .1+ 3.|4!      |
|00003710| 20 20 20 20 20 20 20 20 | 20 20 20 21 11 33 78 20 |        |   !.3x |
|00003720| 11 31 2b 20 32 11 34 21 | 0d 0b 00 20 20 20 20 20 |.1+ 2.4!|...     |
|00003730| 20 20 20 20 20 20 20 20 | 20 39 20 20 20 20 20 30 |        | 9     0|
|00003740| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003750| 20 39 20 20 20 20 20 30 | 0d 0b 00 20 20 20 20 20 | 9     0|...     |
|00003760| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003770| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003780| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 32 0d |        |    .22.|
|00003790| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000037a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000037b0| 20 11 31 34 11 33 78 20 | 11 31 2b 20 38 20 2d 20 | .14.3x |.1+ 8 - |
|000037c0| 11 33 78 20 11 31 2d 20 | 33 20 3d 20 11 33 78 20 |.3x .1- |3 = .3x |
|000037d0| 20 11 31 2b 20 35 11 33 | 78 20 11 31 2b 20 36 2c | .1+ 5.3|x .1+ 6,|
|000037e0| 20 11 33 78 20 11 34 3d | 20 11 31 2d 32 2c 20 2d | .3x .4=| .1-2, -|
|000037f0| 33 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |3...    |        |
|00003800| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003810| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003820| 20 20 20 20 20 11 32 32 | 0d 0b 00 20 20 20 20 20 |     .22|...     |
|00003830| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003840| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003850| 20 20 20 20 20 20 20 11 | 31 30 20 3d 20 11 33 78 |       .|10 = .3x|
|00003860| 20 20 11 31 2b 20 32 11 | 33 78 20 11 31 2b 20 31 |  .1+ 2.|3x .1+ 1|
|00003870| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00003880| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003890| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000038a0| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 0d 0b 00 |        |  .22...|
|000038b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000038c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000038d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 30 20 |        |    .10 |
|000038e0| 3d 20 28 11 33 78 20 11 | 31 2b 20 31 29 0d 0a 00 |= (.3x .|1+ 1)...|
|000038f0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00003900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003910| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 2d 31 |        |      -1|
|00003920| 20 3d 20 11 33 78 0d 0a | 00 00 17 4f 03 0d 0b 00 | = .3x..|...O....|
|00003930| 57 72 6f 6e 67 2e 20 20 | 54 68 65 20 67 69 76 65 |Wrong.  |The give|
|00003940| 6e 20 65 71 75 61 74 69 | 6f 6e 20 68 61 73 20 6f |n equati|on has o|
|00003950| 6e 65 20 72 65 61 6c 20 | 73 6f 6c 75 74 69 6f 6e |ne real |solution|
|00003960| 2e 20 20 54 72 79 20 61 | 67 61 69 6e 2e 0d 0a 00 |.  Try a|gain....|
|00003970| 00 17 4f 08 0d 0b 00 4e | 6f 2e 20 20 53 69 6e 63 |..O....N|o.  Sinc|
|00003980| 65 20 74 68 65 20 76 61 | 72 69 61 62 6c 65 20 65 |e the va|riable e|
|00003990| 78 70 72 65 73 73 69 6f | 6e 20 69 6e 73 69 64 65 |xpressio|n inside|
|000039a0| 20 74 68 65 20 61 62 73 | 6f 6c 75 74 65 20 76 61 | the abs|olute va|
|000039b0| 6c 75 65 20 73 69 67 6e | 73 20 63 61 6e 20 0d 0a |lue sign|s can ..|
|000039c0| 00 62 65 20 70 6f 73 69 | 74 69 76 65 20 6f 72 20 |.be posi|tive or |
|000039d0| 6e 65 67 61 74 69 76 65 | 2c 20 77 65 20 6d 75 73 |negative|, we mus|
|000039e0| 74 20 73 6f 6c 76 65 20 | 74 68 65 20 66 6f 6c 6c |t solve |the foll|
|000039f0| 6f 77 69 6e 67 20 74 77 | 6f 20 65 71 75 61 74 69 |owing tw|o equati|
|00003a00| 6f 6e 73 2e 12 30 0d 0a | 00 20 20 20 20 20 46 69 |ons..0..|.     Fi|
|00003a10| 72 73 74 20 65 71 75 61 | 74 69 6f 6e 3a 20 34 11 |rst equa|tion: 4.|
|00003a20| 33 78 20 11 31 2d 20 31 | 20 3d 20 37 20 20 20 20 |3x .1- 1| = 7    |
|00003a30| 20 53 65 63 6f 6e 64 20 | 65 71 75 61 74 69 6f 6e | Second |equation|
|00003a40| 3a 20 2d 28 34 11 33 78 | 20 11 31 2d 20 31 29 20 |: -(4.3x| .1- 1) |
|00003a50| 3d 20 37 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |= 7...  |        |
|00003a60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 34 |        |       4|
|00003a70| 11 33 78 20 11 31 3d 20 | 38 20 20 20 20 20 20 20 |.3x .1= |8       |
|00003a80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003a90| 20 2d 34 11 33 78 20 11 | 31 2b 20 31 20 3d 20 37 | -4.3x .|1+ 1 = 7|
|00003aa0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00003ab0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |        |     .3x|
|00003ac0| 20 11 31 3d 20 32 20 20 | 20 20 20 20 20 20 20 20 | .1= 2  |        |
|00003ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ae0| 20 20 2d 34 11 33 78 20 | 11 31 3d 20 36 0d 0a 00 |  -4.3x |.1= 6...|
|00003af0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003b00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003b10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003b20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |        |     .3x|
|00003b30| 20 11 31 3d 20 2d 33 2f | 32 0d 0a 00 00 17 4f 03 | .1= -3/|2.....O.|
|00003b40| 0d 0b 00 56 65 72 79 20 | 67 6f 6f 64 2e 20 20 54 |...Very |good.  T|
|00003b50| 72 79 20 63 68 65 63 6b | 69 6e 67 20 74 68 65 73 |ry check|ing thes|
|00003b60| 65 20 73 6f 6c 75 74 69 | 6f 6e 73 20 69 6e 20 74 |e soluti|ons in t|
|00003b70| 68 65 20 6f 72 69 67 69 | 6e 61 6c 20 65 71 75 61 |he origi|nal equa|
|00003b80| 74 69 6f 6e 2e 0d 0a 00 | 00 17 4f 08 0d 0b 00 57 |tion....|..O....W|
|00003b90| 72 6f 6e 67 2e 20 20 53 | 69 6e 63 65 20 74 68 65 |rong.  S|ince the|
|00003ba0| 20 76 61 72 69 61 62 6c | 65 20 65 78 70 72 65 73 | variabl|e expres|
|00003bb0| 73 69 6f 6e 20 69 6e 73 | 69 64 65 20 74 68 65 20 |sion ins|ide the |
|00003bc0| 61 62 73 6f 6c 75 74 65 | 20 76 61 6c 75 65 20 73 |absolute| value s|
|00003bd0| 69 67 6e 73 20 63 61 6e | 20 0d 0a 00 62 65 20 70 |igns can| ...be p|
|00003be0| 6f 73 69 74 69 76 65 20 | 6f 72 20 6e 65 67 61 74 |ositive |or negat|
|00003bf0| 69 76 65 2c 20 77 65 20 | 6d 75 73 74 20 73 6f 6c |ive, we |must sol|
|00003c00| 76 65 20 74 68 65 20 66 | 6f 6c 6c 6f 77 69 6e 67 |ve the f|ollowing|
|00003c10| 20 74 77 6f 20 65 71 75 | 61 74 69 6f 6e 73 2e 12 | two equ|ations..|
|00003c20| 30 0d 0a 00 20 20 20 20 | 20 46 69 72 73 74 20 65 |0...    | First e|
|00003c30| 71 75 61 74 69 6f 6e 3a | 20 34 11 33 78 20 11 31 |quation:| 4.3x .1|
|00003c40| 2d 20 31 20 3d 20 37 20 | 20 20 20 20 53 65 63 6f |- 1 = 7 |    Seco|
|00003c50| 6e 64 20 65 71 75 61 74 | 69 6f 6e 3a 20 2d 28 34 |nd equat|ion: -(4|
|00003c60| 11 33 78 20 11 31 2d 20 | 31 29 20 3d 20 37 0d 0a |.3x .1- |1) = 7..|
|00003c70| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00003c80| 20 20 20 20 20 20 20 20 | 20 20 34 11 33 78 20 11 |        |  4.3x .|
|00003c90| 31 3d 20 38 20 20 20 20 | 20 20 20 20 20 20 20 20 |1= 8    |        |
|00003ca0| 20 20 20 20 20 20 20 20 | 20 20 20 20 2d 34 11 33 |        |    -4.3|
|00003cb0| 78 20 11 31 2b 20 31 20 | 3d 20 37 0d 0a 00 20 20 |x .1+ 1 |= 7...  |
|00003cc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003cd0| 20 20 20 20 20 20 20 20 | 11 33 78 20 11 31 3d 20 |        |.3x .1= |
|00003ce0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|00003cf0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 2d 34 11 |        |     -4.|
|00003d00| 33 78 20 11 31 3d 20 36 | 0d 0a 00 20 20 20 20 20 |3x .1= 6|...     |
|00003d10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003d20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003d30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003d40| 20 20 20 20 20 20 20 20 | 11 33 78 20 11 31 3d 20 |        |.3x .1= |
|00003d50| 2d 33 2f 32 0d 0a 00 00 | 17 4f 08 0d 0b 00 57 72 |-3/2....|.O....Wr|
|00003d60| 6f 6e 67 2e 20 20 53 69 | 6e 63 65 20 74 68 65 20 |ong.  Si|nce the |
|00003d70| 76 61 72 69 61 62 6c 65 | 20 65 78 70 72 65 73 73 |variable| express|
|00003d80| 69 6f 6e 20 69 6e 73 69 | 64 65 20 74 68 65 20 61 |ion insi|de the a|
|00003d90| 62 73 6f 6c 75 74 65 20 | 76 61 6c 75 65 20 73 69 |bsolute |value si|
|00003da0| 67 6e 73 20 63 61 6e 20 | 0d 0a 00 62 65 20 70 6f |gns can |...be po|
|00003db0| 73 69 74 69 76 65 20 6f | 72 20 6e 65 67 61 74 69 |sitive o|r negati|
|00003dc0| 76 65 2c 20 77 65 20 6d | 75 73 74 20 73 6f 6c 76 |ve, we m|ust solv|
|00003dd0| 65 20 74 68 65 20 66 6f | 6c 6c 6f 77 69 6e 67 20 |e the fo|llowing |
|00003de0| 74 77 6f 20 65 71 75 61 | 74 69 6f 6e 73 2e 12 30 |two equa|tions..0|
|00003df0| 0d 0a 00 20 20 20 20 20 | 46 69 72 73 74 20 65 71 |...     |First eq|
|00003e00| 75 61 74 69 6f 6e 3a 20 | 34 11 33 78 20 11 31 2d |uation: |4.3x .1-|
|00003e10| 20 31 20 3d 20 37 20 20 | 20 20 20 53 65 63 6f 6e | 1 = 7  |   Secon|
|00003e20| 64 20 65 71 75 61 74 69 | 6f 6e 3a 20 2d 28 34 11 |d equati|on: -(4.|
|00003e30| 33 78 20 11 31 2d 20 31 | 29 20 3d 20 37 0d 0a 00 |3x .1- 1|) = 7...|
|00003e40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003e50| 20 20 20 20 20 20 20 20 | 20 34 11 33 78 20 11 31 |        | 4.3x .1|
|00003e60| 3d 20 38 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |= 8     |        |
|00003e70| 20 20 20 20 20 20 20 20 | 20 20 20 2d 34 11 33 78 |        |   -4.3x|
|00003e80| 20 11 31 2b 20 31 20 3d | 20 37 0d 0a 00 20 20 20 | .1+ 1 =| 7...   |
|00003e90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ea0| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 3d 20 32 |       .|3x .1= 2|
|00003eb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ec0| 20 20 20 20 20 20 20 20 | 20 20 20 20 2d 34 11 33 |        |    -4.3|
|00003ed0| 78 20 11 31 3d 20 36 0d | 0a 00 20 20 20 20 20 20 |x .1= 6.|..      |
|00003ee0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ef0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003f00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003f10| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 3d 20 2d |       .|3x .1= -|
|00003f20| 33 2f 32 0d 0a 00 00 17 | 4f 03 0d 0b 00 56 65 72 |3/2.....|O....Ver|
|00003f30| 79 20 67 6f 6f 64 2e 20 | 20 54 72 79 20 63 68 65 |y good. | Try che|
|00003f40| 63 6b 69 6e 67 20 79 6f | 75 72 20 61 6e 73 77 65 |cking yo|ur answe|
|00003f50| 72 73 20 69 6e 20 74 68 | 65 20 6f 72 69 67 69 6e |rs in th|e origin|
|00003f60| 61 6c 20 65 71 75 61 74 | 69 6f 6e 2e 0d 0a 00 00 |al equat|ion.....|
|00003f70| 17 4f 06 0d 0b 00 57 72 | 6f 6e 67 2e 20 20 41 66 |.O....Wr|ong.  Af|
|00003f80| 74 65 72 20 73 6f 6c 76 | 69 6e 67 20 74 68 65 20 |ter solv|ing the |
|00003f90| 67 69 76 65 6e 20 65 71 | 75 61 74 69 6f 6e 2c 20 |given eq|uation, |
|00003fa0| 77 65 20 6f 62 74 61 69 | 6e 20 11 33 78 20 11 31 |we obtai|n .3x .1|
|00003fb0| 3d 20 2d 31 2c 20 11 33 | 78 20 11 31 3d 20 31 2c |= -1, .3|x .1= 1,|
|00003fc0| 20 61 6e 64 20 0d 0a 00 | 11 33 78 20 11 31 3d 20 | and ...|.3x .1= |
|00003fd0| 31 32 2e 20 20 43 68 65 | 63 6b 69 6e 67 20 74 68 |12.  Che|cking th|
|00003fe0| 65 73 65 20 74 72 69 61 | 6c 20 73 6f 6c 75 74 69 |ese tria|l soluti|
|00003ff0| 6f 6e 73 20 69 6e 20 74 | 68 65 20 6f 72 69 67 69 |ons in t|he origi|
|00004000| 6e 61 6c 20 65 71 75 61 | 74 69 6f 6e 2c 20 77 65 |nal equa|tion, we|
|00004010| 20 66 69 6e 64 0d 0a 00 | 74 68 61 74 20 11 33 78 | find...|that .3x|
|00004020| 20 11 31 3d 20 31 20 64 | 6f 65 73 20 6e 6f 74 20 | .1= 1 d|oes not |
|00004030| 63 68 65 63 6b 20 28 73 | 65 65 20 62 65 6c 6f 77 |check (s|ee below|
|00004040| 29 2e 12 30 0d 0a 00 12 | 31 20 20 20 20 20 20 20 |)..0....|1       |
|00004050| 20 20 20 20 20 20 20 3f | 12 30 20 20 20 20 11 32 |       ?|.0    .2|
|00004060| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|00004070| 20 20 11 31 12 31 20 20 | 20 20 20 20 3f 20 20 20 |  .1.1  |    ?   |
|00004080| 20 20 20 20 20 20 20 20 | 20 20 20 20 12 30 0d 0b |        |    .0..|
|00004090| 00 20 20 20 20 20 7c 31 | 20 2d 20 31 32 7c 20 3d |.     |1| - 12| =|
|000040a0| 20 28 31 29 20 20 2d 20 | 31 32 28 31 29 20 20 20 | (1)  - |12(1)   |
|000040b0| 11 34 35 35 36 20 20 20 | 11 31 7c 2d 31 32 7c 20 |.4556   |.1|-12| |
|000040c0| 3d 20 31 20 2d 20 31 32 | 20 20 20 11 34 35 35 36 |= 1 - 12|   .4556|
|000040d0| 20 20 20 11 31 31 32 20 | 11 34 3d 20 11 31 2d 31 |   .112 |.4= .1-1|
|000040e0| 31 0d 0a 00 00 17 4f 06 | 0d 0b 00 57 72 6f 6e 67 |1.....O.|...Wrong|
|000040f0| 2e 20 20 41 66 74 65 72 | 20 73 6f 6c 76 69 6e 67 |.  After| solving|
|00004100| 20 74 68 65 20 67 69 76 | 65 6e 20 65 71 75 61 74 | the giv|en equat|
|00004110| 69 6f 6e 2c 20 77 65 20 | 6f 62 74 61 69 6e 20 11 |ion, we |obtain .|
|00004120| 33 78 20 11 31 3d 20 2d | 31 2c 20 11 33 78 20 11 |3x .1= -|1, .3x .|
|00004130| 31 3d 20 31 2c 20 61 6e | 64 20 0d 0a 00 11 33 78 |1= 1, an|d ....3x|
|00004140| 20 11 31 3d 20 31 32 2e | 20 20 43 68 65 63 6b 69 | .1= 12.|  Checki|
|00004150| 6e 67 20 74 68 65 73 65 | 20 74 72 69 61 6c 20 73 |ng these| trial s|
|00004160| 6f 6c 75 74 69 6f 6e 73 | 20 69 6e 20 74 68 65 20 |olutions| in the |
|00004170| 6f 72 69 67 69 6e 61 6c | 20 65 71 75 61 74 69 6f |original| equatio|
|00004180| 6e 2c 20 77 65 20 66 69 | 6e 64 0d 0a 00 74 68 61 |n, we fi|nd...tha|
|00004190| 74 20 11 33 78 20 11 31 | 3d 20 31 20 64 6f 65 73 |t .3x .1|= 1 does|
|000041a0| 20 6e 6f 74 20 63 68 65 | 63 6b 20 28 73 65 65 20 | not che|ck (see |
|000041b0| 62 65 6c 6f 77 29 2e 12 | 30 0d 0a 00 12 31 20 20 |below)..|0....1  |
|000041c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 3f 12 30 20 |        |    ?.0 |
|000041d0| 20 20 20 11 32 32 20 20 | 20 20 20 20 20 20 20 20 |   .22  |        |
|000041e0| 20 20 20 20 20 20 20 11 | 31 12 31 20 20 20 20 20 |       .|1.1     |
|000041f0| 20 3f 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 | ?      |        |
|00004200| 20 12 30 0d 0b 00 20 20 | 20 20 20 7c 31 20 2d 20 | .0...  |   |1 - |
|00004210| 31 32 7c 20 3d 20 28 31 | 29 20 20 2d 20 31 32 28 |12| = (1|)  - 12(|
|00004220| 31 29 20 20 20 11 34 35 | 35 36 20 20 20 11 31 7c |1)   .45|56   .1||
|00004230| 2d 31 32 7c 20 3d 20 31 | 20 2d 20 31 32 20 20 20 |-12| = 1| - 12   |
|00004240| 11 34 35 35 36 20 20 20 | 11 31 31 32 20 11 34 3d |.4556   |.112 .4=|
|00004250| 20 11 31 2d 31 31 0d 0a | 00 00 17 4f 03 0d 0b 00 | .1-11..|...O....|
|00004260| 57 72 6f 6e 67 2e 20 20 | 54 68 65 20 67 69 76 65 |Wrong.  |The give|
|00004270| 6e 20 65 71 75 61 74 69 | 6f 6e 20 68 61 73 20 74 |n equati|on has t|
|00004280| 77 6f 20 72 65 61 6c 20 | 73 6f 6c 75 74 69 6f 6e |wo real |solution|
|00004290| 73 2e 20 20 54 72 79 20 | 61 67 61 69 6e 2e 0d 0a |s.  Try |again...|
|000042a0| 00 00 17 4f 06 0d 0b 00 | 4e 6f 2e 20 20 4f 6e 65 |...O....|No.  One|
|000042b0| 20 6f 66 20 74 68 65 20 | 73 6f 6c 75 74 69 6f 6e | of the |solution|
|000042c0| 73 20 69 73 20 35 2c 20 | 62 75 74 20 74 68 65 72 |s is 5, |but ther|
|000042d0| 65 20 61 72 65 20 74 77 | 6f 20 6f 74 68 65 72 20 |e are tw|o other |
|000042e0| 72 65 61 6c 20 73 6f 6c | 75 74 69 6f 6e 73 2e 20 |real sol|utions. |
|000042f0| 20 0d 0a 00 20 20 20 20 | 20 20 20 11 32 33 20 20 | ...    |   .23  |
|00004300| 20 20 20 32 0d 0b 00 11 | 31 53 69 6e 63 65 20 11 |   2....|1Since .|
|00004310| 33 78 20 20 11 31 2d 20 | 35 11 33 78 20 20 11 31 |3x  .1- |5.3x  .1|
|00004320| 2d 20 11 33 78 20 11 31 | 2b 20 35 20 3d 20 30 20 |- .3x .1|+ 5 = 0 |
|00004330| 63 61 6e 20 62 65 20 66 | 61 63 74 6f 72 65 64 20 |can be f|actored |
|00004340| 61 73 20 28 11 33 78 20 | 11 31 2d 20 35 29 28 11 |as (.3x |.1- 5)(.|
|00004350| 33 78 20 11 31 2d 20 31 | 29 28 11 33 78 20 11 31 |3x .1- 1|)(.3x .1|
|00004360| 2b 20 31 29 20 3d 20 30 | 2c 0d 0a 00 0d 0b 00 77 |+ 1) = 0|,......w|
|00004370| 65 20 63 61 6e 20 73 65 | 65 20 74 68 61 74 20 74 |e can se|e that t|
|00004380| 68 65 20 72 65 61 6c 20 | 73 6f 6c 75 74 69 6f 6e |he real |solution|
|00004390| 73 20 61 72 65 20 35 2c | 20 31 2c 20 61 6e 64 20 |s are 5,| 1, and |
|000043a0| 2d 31 2e 0d 0a 00 00 17 | 4f 03 0d 0b 00 47 6f 6f |-1......|O....Goo|
|000043b0| 64 20 6a 6f 62 2e 20 20 | 54 68 65 20 67 69 76 65 |d job.  |The give|
|000043c0| 6e 20 65 71 75 61 74 69 | 6f 6e 20 68 61 73 20 74 |n equati|on has t|
|000043d0| 68 72 65 65 20 73 6f 6c | 75 74 69 6f 6e 73 3a 20 |hree sol|utions: |
|000043e0| 35 2c 20 31 2c 20 61 6e | 64 20 2d 31 2e 0d 0a 00 |5, 1, an|d -1....|
|000043f0| 00 17 4f 06 0d 0b 00 4e | 6f 2e 20 20 4f 6e 65 20 |..O....N|o.  One |
|00004400| 6f 66 20 74 68 65 20 73 | 6f 6c 75 74 69 6f 6e 73 |of the s|olutions|
|00004410| 20 69 73 20 35 2c 20 62 | 75 74 20 74 68 65 72 65 | is 5, b|ut there|
|00004420| 20 61 72 65 20 74 77 6f | 20 6f 74 68 65 72 20 72 | are two| other r|
|00004430| 65 61 6c 20 73 6f 6c 75 | 74 69 6f 6e 73 2e 20 20 |eal solu|tions.  |
|00004440| 0d 0a 00 20 20 20 20 20 | 20 20 11 32 33 20 20 20 |...     |  .23   |
|00004450| 20 20 32 0d 0b 00 11 31 | 53 69 6e 63 65 20 11 33 |  2....1|Since .3|
|00004460| 78 20 20 11 31 2d 20 35 | 11 33 78 20 20 11 31 2d |x  .1- 5|.3x  .1-|
|00004470| 20 11 33 78 20 11 31 2b | 20 35 20 3d 20 30 20 63 | .3x .1+| 5 = 0 c|
|00004480| 61 6e 20 62 65 20 66 61 | 63 74 6f 72 65 64 20 61 |an be fa|ctored a|
|00004490| 73 20 28 11 33 78 20 11 | 31 2d 20 35 29 28 11 33 |s (.3x .|1- 5)(.3|
|000044a0| 78 20 11 31 2d 20 31 29 | 28 11 33 78 20 11 31 2b |x .1- 1)|(.3x .1+|
|000044b0| 20 31 29 20 3d 20 30 2c | 0d 0a 00 0d 0b 00 77 65 | 1) = 0,|......we|
|000044c0| 20 63 61 6e 20 73 65 65 | 20 74 68 61 74 20 74 68 | can see| that th|
|000044d0| 65 20 72 65 61 6c 20 73 | 6f 6c 75 74 69 6f 6e 73 |e real s|olutions|
|000044e0| 20 61 72 65 20 35 2c 20 | 31 2c 20 61 6e 64 20 2d | are 5, |1, and -|
|000044f0| 31 2e 0d 0a 00 00 17 4f | 03 0d 0b 00 57 72 6f 6e |1......O|....Wron|
|00004500| 67 2e 20 20 54 68 65 20 | 67 69 76 65 6e 20 65 71 |g.  The |given eq|
|00004510| 75 61 74 69 6f 6e 20 68 | 61 73 20 74 68 72 65 65 |uation h|as three|
|00004520| 20 72 65 61 6c 20 73 6f | 6c 75 74 69 6f 6e 73 2e | real so|lutions.|
|00004530| 20 20 54 72 79 20 61 67 | 61 69 6e 2e 0d 0a 00 53 |  Try ag|ain....S|
|00004540| 65 63 74 69 6f 6e 20 31 | 2e 36 20 20 4f 74 68 65 |ection 1|.6  Othe|
|00004550| 72 20 54 79 70 65 73 20 | 6f 66 20 45 71 75 61 74 |r Types |of Equat|
|00004560| 69 6f 6e 73 0d 0a 00 0d | 0a 00 0d 0b 00 20 20 20 |ions....|.....   |
|00004570| 20 0e 78 31 2d 36 2d 31 | 0e 54 75 74 6f 72 69 61 | .x1-6-1|.Tutoria|
|00004580| 6c 20 45 78 65 72 63 69 | 73 65 20 20 31 0f 20 20 |l Exerci|se  1.  |
|00004590| 20 20 0e 78 31 2d 36 2d | 32 0e 54 75 74 6f 72 69 |  .x1-6-|2.Tutori|
|000045a0| 61 6c 20 45 78 65 72 63 | 69 73 65 20 20 32 0f 20 |al Exerc|ise  2. |
|000045b0| 20 20 20 0e 78 31 2d 36 | 2d 33 0e 54 75 74 6f 72 |   .x1-6|-3.Tutor|
|000045c0| 69 61 6c 20 45 78 65 72 | 63 69 73 65 20 20 33 0f |ial Exer|cise  3.|
|000045d0| 20 20 20 20 0e 78 31 2d | 36 2d 34 0e 54 75 74 6f |    .x1-|6-4.Tuto|
|000045e0| 72 69 61 6c 20 45 78 65 | 72 63 69 73 65 20 20 34 |rial Exe|rcise  4|
|000045f0| 0f 20 20 20 20 0e 78 31 | 2d 36 2d 35 0e 54 75 74 |.    .x1|-6-5.Tut|
|00004600| 6f 72 69 61 6c 20 45 78 | 65 72 63 69 73 65 20 20 |orial Ex|ercise  |
|00004610| 35 0f 0d 0a 00 0d 0b 00 | 20 20 20 20 0e 78 31 2d |5.......|    .x1-|
|00004620| 36 2d 36 0e 54 75 74 6f | 72 69 61 6c 20 45 78 65 |6-6.Tuto|rial Exe|
|00004630| 72 63 69 73 65 20 20 36 | 0f 20 20 20 20 0e 78 31 |rcise  6|.    .x1|
|00004640| 2d 36 2d 37 0e 54 75 74 | 6f 72 69 61 6c 20 45 78 |-6-7.Tut|orial Ex|
|00004650| 65 72 63 69 73 65 20 20 | 37 0f 0d 0a 00 0d 0b 00 |ercise  |7.......|
|00004660| 20 20 20 20 0e 78 31 2d | 36 2d 38 0e 54 75 74 6f |    .x1-|6-8.Tuto|
|00004670| 72 69 61 6c 20 45 78 65 | 72 63 69 73 65 20 20 38 |rial Exe|rcise  8|
|00004680| 0f 20 20 20 20 0e 78 31 | 2d 36 2d 39 0e 54 75 74 |.    .x1|-6-9.Tut|
|00004690| 6f 72 69 61 6c 20 45 78 | 65 72 63 69 73 65 20 20 |orial Ex|ercise  |
|000046a0| 39 0f 0d 0a 00 0d 0b 00 | 20 20 20 20 0e 78 31 2d |9.......|    .x1-|
|000046b0| 36 2d 31 30 0e 54 75 74 | 6f 72 69 61 6c 20 45 78 |6-10.Tut|orial Ex|
|000046c0| 65 72 63 69 73 65 20 31 | 30 0f 20 20 20 20 0e 78 |ercise 1|0.    .x|
|000046d0| 31 2d 36 2d 31 31 0e 54 | 75 74 6f 72 69 61 6c 20 |1-6-11.T|utorial |
|000046e0| 45 78 65 72 63 69 73 65 | 20 31 31 0f 0d 0a 00 0d |Exercise| 11.....|
|000046f0| 0b 00 20 20 20 20 0e 78 | 31 2d 36 2d 31 32 0e 54 |..    .x|1-6-12.T|
|00004700| 75 74 6f 72 69 61 6c 20 | 45 78 65 72 63 69 73 65 |utorial |Exercise|
|00004710| 20 31 32 0f 0d 0a 00 0d | 0b 00 20 20 20 20 0e 78 | 12.....|..    .x|
|00004720| 31 2d 36 2d 31 33 0e 54 | 75 74 6f 72 69 61 6c 20 |1-6-13.T|utorial |
|00004730| 45 78 65 72 63 69 73 65 | 20 31 33 0f 20 20 20 20 |Exercise| 13.    |
|00004740| 0e 78 31 2d 36 2d 31 34 | 0e 54 75 74 6f 72 69 61 |.x1-6-14|.Tutoria|
|00004750| 6c 20 45 78 65 72 63 69 | 73 65 20 31 34 0f 0d 0a |l Exerci|se 14...|
|00004760| 00 53 65 63 74 69 6f 6e | 20 31 2e 36 20 20 4f 74 |.Section| 1.6  Ot|
|00004770| 68 65 72 20 54 79 70 65 | 73 20 6f 66 20 45 71 75 |her Type|s of Equ|
|00004780| 61 74 69 6f 6e 73 20 20 | 20 20 20 20 20 20 20 20 |ations  |        |
|00004790| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000047a0| 20 20 20 20 20 20 20 11 | 32 34 0d 0b 00 20 11 31 |       .|24... .1|
|000047b0| 31 2e 20 20 46 69 6e 64 | 20 74 68 65 20 72 65 61 |1.  Find| the rea|
|000047c0| 6c 20 73 6f 6c 75 74 69 | 6f 6e 73 20 6f 66 20 11 |l soluti|ons of .|
|000047d0| 33 78 20 20 11 31 2d 20 | 31 36 20 3d 20 30 2e 0d |3x  .1- |16 = 0..|
|000047e0| 0a 00 0d 0a 00 20 20 20 | 20 20 0e 61 32 2d 36 2d |.....   |  .a2-6-|
|000047f0| 35 61 0e 28 61 29 0f 20 | 11 34 2b 11 31 32 20 20 |5a.(a). |.4+.12  |
|00004800| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004810| 20 20 20 20 20 20 20 20 | 20 0e 61 32 2d 36 2d 35 |        | .a2-6-5|
|00004820| 62 0e 28 62 29 0f 20 11 | 34 2b 11 31 34 0d 0a 00 |b.(b). .|4+.14...|
|00004830| 0d 0b 00 20 20 20 20 20 | 0e 61 32 2d 36 2d 35 63 |...     |.a2-6-5c|
|00004840| 0e 28 63 29 0f 20 11 34 | 2b 11 31 32 2c 20 11 34 |.(c). .4|+.12, .4|
|00004850| 2b 11 31 34 20 20 20 20 | 20 20 20 20 20 20 20 20 |+.14    |        |
|00004860| 20 20 20 20 20 20 20 20 | 20 20 20 0e 61 32 2d 36 |        |   .a2-6|
|00004870| 2d 35 64 0e 28 64 29 0f | 20 4e 6f 20 72 65 61 6c |-5d.(d).| No real|
|00004880| 20 73 6f 6c 75 74 69 6f | 6e 0d 0a 00 0d 0a 00 59 | solutio|n......Y|
|00004890| 6f 75 20 6d 61 79 20 77 | 61 6e 74 20 74 6f 20 72 |ou may w|ant to r|
|000048a0| 65 76 69 65 77 20 10 31 | 2d 36 2d 32 0e 65 31 2d |eview .1|-6-2.e1-|
|000048b0| 36 2d 31 0e 47 75 69 64 | 65 64 20 45 78 61 6d 70 |6-1.Guid|ed Examp|
|000048c0| 6c 65 20 31 2e 0f 0d 0a | 00 53 65 63 74 69 6f 6e |le 1....|.Section|
|000048d0| 20 31 2e 36 20 20 4f 74 | 68 65 72 20 54 79 70 65 | 1.6  Ot|her Type|
|000048e0| 73 20 6f 66 20 45 71 75 | 61 74 69 6f 6e 73 0d 0b |s of Equ|ations..|
|000048f0| 00 31 30 2e 20 20 46 69 | 6e 64 20 74 68 65 20 72 |.10.  Fi|nd the r|
|00004900| 65 61 6c 20 73 6f 6c 75 | 74 69 6f 6e 73 20 6f 66 |eal solu|tions of|
|00004910| 20 7c 34 11 33 78 20 11 | 31 2d 20 31 7c 20 3d 20 | |4.3x .|1- 1| = |
|00004920| 37 2e 0d 0a 00 0d 0a 00 | 20 20 20 20 20 0e 61 32 |7.......|     .a2|
|00004930| 2d 36 2d 36 35 61 0e 28 | 61 29 0f 20 38 2c 20 2d |-6-65a.(|a). 8, -|
|00004940| 36 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |6       |        |
|00004950| 20 20 20 20 20 20 20 20 | 20 0e 61 32 2d 36 2d 36 |        | .a2-6-6|
|00004960| 35 62 0e 28 62 29 0f 20 | 32 2c 20 2d 33 2f 32 0d |5b.(b). |2, -3/2.|
|00004970| 0a 00 0d 0b 00 20 20 20 | 20 20 0e 61 32 2d 36 2d |.....   |  .a2-6-|
|00004980| 36 35 63 0e 28 63 29 0f | 20 2d 32 20 20 20 20 20 |65c.(c).| -2     |
|00004990| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000049a0| 20 20 20 20 20 20 0e 61 | 32 2d 36 2d 36 35 64 0e |      .a|2-6-65d.|
|000049b0| 28 64 29 0f 20 33 2f 32 | 0d 0a 00 0d 0a 00 59 6f |(d). 3/2|......Yo|
|000049c0| 75 20 6d 61 79 20 77 61 | 6e 74 20 74 6f 20 72 65 |u may wa|nt to re|
|000049d0| 76 69 65 77 20 10 31 2d | 36 2d 32 0e 65 31 2d 36 |view .1-|6-2.e1-6|
|000049e0| 2d 38 0e 47 75 69 64 65 | 64 20 45 78 61 6d 70 6c |-8.Guide|d Exampl|
|000049f0| 65 20 38 2e 0f 0d 0a 00 | 53 65 63 74 69 6f 6e 20 |e 8.....|Section |
|00004a00| 31 2e 36 20 20 4f 74 68 | 65 72 20 54 79 70 65 73 |1.6  Oth|er Types|
|00004a10| 20 6f 66 20 45 71 75 61 | 74 69 6f 6e 73 20 20 20 | of Equa|tions   |
|00004a20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004a30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004a40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004a50| 20 11 32 32 0d 0b 00 11 | 31 31 31 2e 20 20 46 69 | .22....|111.  Fi|
|00004a60| 6e 64 20 61 6c 6c 20 73 | 6f 6c 75 74 69 6f 6e 73 |nd all s|olutions|
|00004a70| 20 6f 66 20 74 68 65 20 | 65 71 75 61 74 69 6f 6e | of the |equation|
|00004a80| 20 7c 11 33 78 20 11 31 | 2d 20 31 32 7c 20 3d 20 | |.3x .1|- 12| = |
|00004a90| 11 33 78 20 20 11 31 2d | 20 31 32 11 33 78 11 31 |.3x  .1-| 12.3x.1|
|00004aa0| 2e 0d 0a 00 0d 0a 00 20 | 20 20 20 20 0e 61 32 2d |....... |    .a2-|
|00004ab0| 36 2d 36 39 61 0e 28 61 | 29 0f 20 2d 31 2c 20 31 |6-69a.(a|). -1, 1|
|00004ac0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|00004ad0| 20 20 20 20 20 20 20 20 | 0e 61 32 2d 36 2d 36 39 |        |.a2-6-69|
|00004ae0| 62 0e 28 62 29 0f 20 2d | 31 2c 20 31 2c 20 31 32 |b.(b). -|1, 1, 12|
|00004af0| 0d 0a 00 0d 0b 00 20 20 | 20 20 20 0e 61 32 2d 36 |......  |   .a2-6|
|00004b00| 2d 36 39 63 0e 28 63 29 | 0f 20 31 2c 20 31 32 20 |-69c.(c)|. 1, 12 |
|00004b10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004b20| 20 20 20 20 20 20 20 0e | 61 32 2d 36 2d 36 39 64 |       .|a2-6-69d|
|00004b30| 0e 28 64 29 0f 20 4e 6f | 20 72 65 61 6c 20 73 6f |.(d). No| real so|
|00004b40| 6c 75 74 69 6f 6e 0d 0a | 00 0d 0a 00 59 6f 75 20 |lution..|....You |
|00004b50| 6d 61 79 20 77 61 6e 74 | 20 74 6f 20 72 65 76 69 |may want| to revi|
|00004b60| 65 77 20 10 31 2d 36 2d | 32 0e 65 31 2d 36 2d 38 |ew .1-6-|2.e1-6-8|
|00004b70| 0e 47 75 69 64 65 64 20 | 45 78 61 6d 70 6c 65 20 |.Guided |Example |
|00004b80| 38 2e 0f 0d 0a 00 53 65 | 63 74 69 6f 6e 20 31 2e |8.....Se|ction 1.|
|00004b90| 36 20 20 4f 74 68 65 72 | 20 54 79 70 65 73 20 6f |6  Other| Types o|
|00004ba0| 66 20 45 71 75 61 74 69 | 6f 6e 73 0d 0b 00 31 32 |f Equati|ons...12|
|00004bb0| 2e 20 20 54 68 65 20 64 | 65 6d 61 6e 64 20 65 71 |.  The d|emand eq|
|00004bc0| 75 61 74 69 6f 6e 20 66 | 6f 72 20 61 20 63 65 72 |uation f|or a cer|
|00004bd0| 74 61 69 6e 20 70 72 6f | 64 75 63 74 20 69 73 20 |tain pro|duct is |
|00004be0| 67 69 76 65 6e 20 62 79 | 20 0d 0a 00 20 20 20 20 |given by| ...    |
|00004bf0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004c00| 20 11 34 44 32 32 32 32 | 32 32 32 32 32 32 0d 0b | .4D2222|222222..|
|00004c10| 00 20 20 20 20 20 20 20 | 20 20 20 20 11 33 70 20 |.       |    .3p |
|00004c20| 11 31 3d 20 35 31 20 2d | 20 11 34 53 20 11 31 30 |.1= 51 -| .4S .10|
|00004c30| 2e 30 30 31 11 33 78 20 | 11 31 2b 20 31 2c 20 30 |.001.3x |.1+ 1, 0|
|00004c40| 20 11 34 3c 20 11 33 78 | 0d 0a 00 0d 0b 00 20 20 | .4< .3x|......  |
|00004c50| 20 20 20 11 31 77 68 65 | 72 65 20 11 33 78 20 11 |   .1whe|re .3x .|
|00004c60| 31 69 73 20 74 68 65 20 | 6e 75 6d 62 65 72 20 6f |1is the |number o|
|00004c70| 66 20 75 6e 69 74 73 20 | 64 65 6d 61 6e 64 65 64 |f units |demanded|
|00004c80| 20 70 65 72 20 64 61 79 | 20 61 6e 64 20 11 33 70 | per day| and .3p|
|00004c90| 20 11 31 69 73 20 74 68 | 65 20 70 72 69 63 65 20 | .1is th|e price |
|00004ca0| 70 65 72 0d 0a 00 20 20 | 20 20 20 75 6e 69 74 2e |per...  |   unit.|
|00004cb0| 20 20 46 69 6e 64 20 74 | 68 65 20 64 65 6d 61 6e |  Find t|he deman|
|00004cc0| 64 20 69 66 20 74 68 65 | 20 70 72 69 63 65 20 70 |d if the| price p|
|00004cd0| 65 72 20 75 6e 69 74 20 | 69 73 20 73 65 74 20 61 |er unit |is set a|
|00004ce0| 74 20 24 33 34 2e 37 30 | 2e 0d 0a 00 0d 0a 00 20 |t $34.70|....... |
|00004cf0| 20 20 20 20 0e 32 2d 36 | 2d 38 33 61 0e 28 61 29 |    .2-6|-83a.(a)|
|00004d00| 0f 20 35 31 20 75 6e 69 | 74 73 20 20 20 20 20 20 |. 51 uni|ts      |
|00004d10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 0e |        |       .|
|00004d20| 32 2d 36 2d 38 33 62 0e | 28 62 29 0f 20 32 36 34 |2-6-83b.|(b). 264|
|00004d30| 2c 36 39 30 20 75 6e 69 | 74 73 0d 0a 00 0d 0b 00 |,690 uni|ts......|
|00004d40| 20 20 20 20 20 0e 32 2d | 36 2d 38 33 63 0e 28 63 |     .2-|6-83c.(c|
|00004d50| 29 0f 20 32 36 36 2c 36 | 39 30 20 75 6e 69 74 73 |). 266,6|90 units|
|00004d60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004d70| 0e 32 2d 36 2d 38 33 64 | 0e 28 64 29 0f 20 37 2c |.2-6-83d|.(d). 7,|
|00004d80| 33 34 34 2c 34 39 30 20 | 75 6e 69 74 73 0d 0a 00 |344,490 |units...|
|00004d90| 0d 0a 00 59 6f 75 20 6d | 61 79 20 77 61 6e 74 20 |...You m|ay want |
|00004da0| 74 6f 20 72 65 76 69 65 | 77 20 10 31 2d 36 2d 32 |to revie|w .1-6-2|
|00004db0| 0e 65 31 2d 36 2d 39 0e | 47 75 69 64 65 64 20 45 |.e1-6-9.|Guided E|
|00004dc0| 78 61 6d 70 6c 65 20 39 | 2e 0f 0d 0a 00 53 65 63 |xample 9|.....Sec|
|00004dd0| 74 69 6f 6e 20 31 2e 36 | 20 20 4f 74 68 65 72 20 |tion 1.6|  Other |
|00004de0| 54 79 70 65 73 20 6f 66 | 20 45 71 75 61 74 69 6f |Types of| Equatio|
|00004df0| 6e 73 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |ns      |       .|
|00004e00| 34 44 32 32 32 32 32 32 | 0d 0b 00 11 31 31 33 2e |4D222222|....113.|
|00004e10| 20 20 53 6f 6c 76 65 3a | 20 11 34 53 20 11 31 38 |  Solve:| .4S .18|
|00004e20| 11 33 78 20 11 31 2b 20 | 31 20 2d 20 32 20 3d 20 |.3x .1+ |1 - 2 = |
|00004e30| 11 33 78 11 31 2e 0d 0a | 00 0d 0a 00 20 20 20 20 |.3x.1...|....    |
|00004e40| 20 0e 32 2d 36 2d 33 35 | 61 0e 28 61 29 0f 20 2d | .2-6-35|a.(a). -|
|00004e50| 31 2c 20 2d 33 20 20 20 | 20 20 20 20 20 20 20 20 |1, -3   |        |
|00004e60| 20 20 20 20 20 20 20 20 | 20 20 20 20 0e 32 2d 36 |        |    .2-6|
|00004e70| 2d 33 35 62 0e 28 62 29 | 0f 20 31 2f 37 0d 0a 00 |-35b.(b)|. 1/7...|
|00004e80| 0d 0b 00 20 20 20 20 20 | 0e 32 2d 36 2d 33 35 63 |...     |.2-6-35c|
|00004e90| 0e 28 63 29 0f 20 31 2c | 20 33 20 20 20 20 20 20 |.(c). 1,| 3      |
|00004ea0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004eb0| 20 20 20 0e 32 2d 36 2d | 33 35 64 0e 28 64 29 0f |   .2-6-|35d.(d).|
|00004ec0| 20 4e 6f 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f | No real| solutio|
|00004ed0| 6e 0d 0a 00 0d 0a 00 59 | 6f 75 20 6d 61 79 20 77 |n......Y|ou may w|
|00004ee0| 61 6e 74 20 74 6f 20 72 | 65 76 69 65 77 20 10 31 |ant to r|eview .1|
|00004ef0| 2d 36 2d 32 0e 69 31 2d | 36 2d 31 0e 49 6e 74 65 |-6-2.i1-|6-1.Inte|
|00004f00| 67 72 61 74 65 64 20 45 | 78 61 6d 70 6c 65 20 31 |grated E|xample 1|
|00004f10| 2e 0f 0d 0a 00 53 65 63 | 74 69 6f 6e 20 31 2e 36 |.....Sec|tion 1.6|
|00004f20| 20 20 4f 74 68 65 72 20 | 54 79 70 65 73 20 6f 66 |  Other |Types of|
|00004f30| 20 45 71 75 61 74 69 6f | 6e 73 20 20 20 20 20 20 | Equatio|ns      |
|00004f40| 20 20 20 20 20 20 20 11 | 34 44 32 32 20 20 20 20 |       .|4D22    |
|00004f50| 44 32 32 32 32 32 32 0d | 0b 00 11 31 31 34 2e 20 |D222222.|...114. |
|00004f60| 20 53 6f 6c 76 65 3a 20 | 11 34 53 20 11 31 32 11 | Solve: |.4S .12.|
|00004f70| 33 74 20 11 31 2b 20 11 | 34 53 20 11 31 32 11 33 |3t .1+ .|4S .12.3|
|00004f80| 74 20 11 31 2b 20 33 20 | 3d 20 33 2e 0d 0a 00 0d |t .1+ 3 |= 3.....|
|00004f90| 0a 00 20 20 20 20 20 0e | 32 2d 36 2d 33 39 61 0e |..     .|2-6-39a.|
|00004fa0| 28 61 29 0f 20 32 20 20 | 20 20 20 20 20 20 20 20 |(a). 2  |        |
|00004fb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004fc0| 20 20 0e 32 2d 36 2d 33 | 39 62 0e 28 62 29 0f 20 |  .2-6-3|9b.(b). |
|00004fd0| 31 2f 32 0d 0a 00 0d 0b | 00 20 20 20 20 20 0e 32 |1/2.....|.     .2|
|00004fe0| 2d 36 2d 33 39 63 0e 28 | 63 29 0f 20 30 20 20 20 |-6-39c.(|c). 0   |
|00004ff0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005000| 20 20 20 20 20 20 20 20 | 20 0e 32 2d 36 2d 33 39 |        | .2-6-39|
|00005010| 64 0e 28 64 29 0f 20 4e | 6f 20 72 65 61 6c 20 73 |d.(d). N|o real s|
|00005020| 6f 6c 75 74 69 6f 6e 0d | 0a 00 0d 0a 00 59 6f 75 |olution.|.....You|
|00005030| 20 6d 61 79 20 77 61 6e | 74 20 74 6f 20 72 65 76 | may wan|t to rev|
|00005040| 69 65 77 20 10 31 2d 36 | 2d 32 0e 69 31 2d 36 2d |iew .1-6|-2.i1-6-|
|00005050| 32 0e 49 6e 74 65 67 72 | 61 74 65 64 20 45 78 61 |2.Integr|ated Exa|
|00005060| 6d 70 6c 65 20 32 2e 0f | 0d 0a 00 53 65 63 74 69 |mple 2..|...Secti|
|00005070| 6f 6e 20 31 2e 36 20 20 | 4f 74 68 65 72 20 54 79 |on 1.6  |Other Ty|
|00005080| 70 65 73 20 6f 66 20 45 | 71 75 61 74 69 6f 6e 73 |pes of E|quations|
|00005090| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000050a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000050b0| 20 11 32 33 20 20 20 20 | 20 32 0d 0b 00 20 11 31 | .23    | 2... .1|
|000050c0| 32 2e 20 20 46 69 6e 64 | 20 74 68 65 20 72 65 61 |2.  Find| the rea|
|000050d0| 6c 20 73 6f 6c 75 74 69 | 6f 6e 73 20 6f 66 20 11 |l soluti|ons of .|
|000050e0| 33 78 20 20 11 31 2d 20 | 35 11 33 78 20 20 11 31 |3x  .1- |5.3x  .1|
|000050f0| 2d 20 11 33 78 20 11 31 | 2b 20 35 20 3d 20 30 2e |- .3x .1|+ 5 = 0.|
|00005100| 0d 0a 00 0d 0a 00 20 20 | 20 20 20 0e 61 32 2d 36 |......  |   .a2-6|
|00005110| 2d 39 61 0e 28 61 29 0f | 20 35 20 20 20 20 20 20 |-9a.(a).| 5      |
|00005120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005130| 20 20 20 20 20 20 0e 61 | 32 2d 36 2d 39 62 0e 28 |      .a|2-6-9b.(|
|00005140| 62 29 0f 20 35 2c 20 11 | 34 2b 11 31 31 0d 0a 00 |b). 5, .|4+.11...|
|00005150| 0d 0b 00 20 20 20 20 20 | 0e 61 32 2d 36 2d 39 63 |...     |.a2-6-9c|
|00005160| 0e 28 63 29 0f 20 30 2c | 20 11 34 2b 11 31 35 20 |.(c). 0,| .4+.15 |
|00005170| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005180| 20 20 20 20 20 20 20 0e | 61 32 2d 36 2d 39 64 0e |       .|a2-6-9d.|
|00005190| 28 64 29 0f 20 4e 6f 20 | 72 65 61 6c 20 73 6f 6c |(d). No |real sol|
|000051a0| 75 74 69 6f 6e 0d 0a 00 | 0d 0a 00 59 6f 75 20 6d |ution...|...You m|
|000051b0| 61 79 20 77 61 6e 74 20 | 74 6f 20 72 65 76 69 65 |ay want |to revie|
|000051c0| 77 20 10 31 2d 36 2d 32 | 0e 65 31 2d 36 2d 32 0e |w .1-6-2|.e1-6-2.|
|000051d0| 47 75 69 64 65 64 20 45 | 78 61 6d 70 6c 65 20 32 |Guided E|xample 2|
|000051e0| 2e 0f 0d 0a 00 53 65 63 | 74 69 6f 6e 20 31 2e 36 |.....Sec|tion 1.6|
|000051f0| 20 20 4f 74 68 65 72 20 | 54 79 70 65 73 20 6f 66 |  Other |Types of|
|00005200| 20 45 71 75 61 74 69 6f | 6e 73 20 20 20 20 20 20 | Equatio|ns      |
|00005210| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005220| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 34 20 20 |        |   .24  |
|00005230| 20 20 20 32 0d 0b 00 20 | 11 31 33 2e 20 20 46 69 |   2... |.13.  Fi|
|00005240| 6e 64 20 74 68 65 20 72 | 65 61 6c 20 73 6f 6c 75 |nd the r|eal solu|
|00005250| 74 69 6f 6e 73 20 6f 66 | 20 11 33 78 20 20 11 31 |tions of| .3x  .1|
|00005260| 2d 20 37 11 33 78 20 20 | 11 31 2d 20 31 38 20 3d |- 7.3x  |.1- 18 =|
|00005270| 20 30 2e 0d 0a 00 0d 0a | 00 20 20 20 20 20 0e 61 | 0......|.     .a|
|00005280| 32 2d 36 2d 31 35 61 0e | 28 61 29 0f 20 11 34 2b |2-6-15a.|(a). .4+|
|00005290| 11 31 32 2c 20 11 34 2b | 11 31 33 20 20 20 20 20 |.12, .4+|.13     |
|000052a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000052b0| 20 20 0e 61 32 2d 36 2d | 31 35 62 0e 28 62 29 0f |  .a2-6-|15b.(b).|
|000052c0| 20 11 34 2b 11 31 33 0d | 0a 00 0d 0b 00 20 20 20 | .4+.13.|.....   |
|000052d0| 20 20 0e 61 32 2d 36 2d | 31 35 63 0e 28 63 29 0f |  .a2-6-|15c.(c).|
|000052e0| 20 11 34 2b 11 31 33 2c | 20 2d 32 20 20 20 20 20 | .4+.13,| -2     |
|000052f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005300| 20 20 0e 61 32 2d 36 2d | 31 35 64 0e 28 64 29 0f |  .a2-6-|15d.(d).|
|00005310| 20 4e 6f 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f | No real| solutio|
|00005320| 6e 0d 0a 00 0d 0a 00 59 | 6f 75 20 6d 61 79 20 77 |n......Y|ou may w|
|00005330| 61 6e 74 20 74 6f 20 72 | 65 76 69 65 77 20 10 31 |ant to r|eview .1|
|00005340| 2d 36 2d 32 0e 65 31 2d | 36 2d 33 0e 47 75 69 64 |-6-2.e1-|6-3.Guid|
|00005350| 65 64 20 45 78 61 6d 70 | 6c 65 20 33 2e 0f 0d 0a |ed Examp|le 3....|
|00005360| 00 53 65 63 74 69 6f 6e | 20 31 2e 36 20 20 4f 74 |.Section| 1.6  Ot|
|00005370| 68 65 72 20 54 79 70 65 | 73 20 6f 66 20 45 71 75 |her Type|s of Equ|
|00005380| 61 74 69 6f 6e 73 20 20 | 20 20 20 20 20 20 20 20 |ations  |        |
|00005390| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000053a0| 20 20 20 20 20 20 20 11 | 32 36 20 20 20 20 20 20 |       .|26      |
|000053b0| 33 0d 0b 00 20 11 31 34 | 2e 20 20 46 69 6e 64 20 |3... .14|.  Find |
|000053c0| 74 68 65 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f |the real| solutio|
|000053d0| 6e 73 20 6f 66 20 11 33 | 78 20 20 11 31 2b 20 32 |ns of .3|x  .1+ 2|
|000053e0| 36 11 33 78 20 20 11 31 | 2d 20 32 37 20 3d 20 30 |6.3x  .1|- 27 = 0|
|000053f0| 2e 0d 0a 00 0d 0a 00 20 | 20 20 20 20 0e 61 32 2d |....... |    .a2-|
|00005400| 36 2d 31 39 61 0e 28 61 | 29 0f 20 33 2c 20 2d 31 |6-19a.(a|). 3, -1|
|00005410| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005420| 20 20 20 20 20 20 20 20 | 0e 61 32 2d 36 2d 31 39 |        |.a2-6-19|
|00005430| 62 0e 28 62 29 0f 20 31 | 2c 20 2d 32 37 0d 0a 00 |b.(b). 1|, -27...|
|00005440| 0d 0b 00 20 20 20 20 20 | 0e 61 32 2d 36 2d 31 39 |...     |.a2-6-19|
|00005450| 63 0e 28 63 29 0f 20 31 | 2c 20 2d 33 20 20 20 20 |c.(c). 1|, -3    |
|00005460| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005470| 20 20 20 20 0e 61 32 2d | 36 2d 31 39 64 0e 28 64 |    .a2-|6-19d.(d|
|00005480| 29 0f 20 4e 6f 20 72 65 | 61 6c 20 73 6f 6c 75 74 |). No re|al solut|
|00005490| 69 6f 6e 0d 0a 00 0d 0a | 00 59 6f 75 20 6d 61 79 |ion.....|.You may|
|000054a0| 20 77 61 6e 74 20 74 6f | 20 72 65 76 69 65 77 20 | want to| review |
|000054b0| 10 31 2d 36 2d 32 0e 65 | 31 2d 36 2d 33 0e 47 75 |.1-6-2.e|1-6-3.Gu|
|000054c0| 69 64 65 64 20 45 78 61 | 6d 70 6c 65 20 33 2e 0f |ided Exa|mple 3..|
|000054d0| 0d 0a 00 53 65 63 74 69 | 6f 6e 20 31 2e 36 20 20 |...Secti|on 1.6  |
|000054e0| 4f 74 68 65 72 20 54 79 | 70 65 73 20 6f 66 20 45 |Other Ty|pes of E|
|000054f0| 71 75 61 74 69 6f 6e 73 | 20 20 20 20 20 20 20 20 |quations|        |
|00005500| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005510| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00005520| 32 2f 33 20 20 20 20 20 | 31 2f 33 0d 0b 00 20 11 |2/3     |1/3... .|
|00005530| 31 35 2e 20 20 46 69 6e | 64 20 61 6c 6c 20 74 68 |15.  Fin|d all th|
|00005540| 65 20 72 65 61 6c 20 73 | 6f 6c 75 74 69 6f 6e 73 |e real s|olutions|
|00005550| 20 6f 66 20 33 11 33 78 | 20 20 20 20 11 31 2b 20 | of 3.3x|    .1+ |
|00005560| 38 11 33 78 20 20 20 20 | 11 31 2d 20 33 20 3d 20 |8.3x    |.1- 3 = |
|00005570| 30 2e 0d 0a 00 0d 0a 00 | 20 20 20 20 20 0e 32 2d |0.......|     .2-|
|00005580| 36 2d 32 35 61 0e 28 61 | 29 0f 20 31 2c 20 33 20 |6-25a.(a|). 1, 3 |
|00005590| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000055a0| 20 20 20 20 20 20 20 20 | 0e 32 2d 36 2d 32 35 62 |        |.2-6-25b|
|000055b0| 0e 28 62 29 0f 20 31 2c | 20 32 37 0d 0a 00 0d 0b |.(b). 1,| 27.....|
|000055c0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 31 20 |.       |      1 |
|000055d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000055f0| 20 31 0d 0b 00 20 20 20 | 20 20 0e 32 2d 36 2d 32 | 1...   |  .2-6-2|
|00005600| 35 63 0e 28 63 29 0f 20 | 2d 33 2c 20 11 34 32 20 |5c.(c). |-3, .42 |
|00005610| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005620| 20 20 20 20 20 20 20 11 | 31 0e 32 2d 36 2d 32 35 |       .|1.2-6-25|
|00005630| 64 0e 28 64 29 0f 20 2d | 32 37 2c 20 11 34 32 32 |d.(d). -|27, .422|
|00005640| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00005650| 11 31 33 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.13     |        |
|00005660| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005670| 20 20 20 20 32 37 0d 0a | 00 0d 0a 00 59 6f 75 20 |    27..|....You |
|00005680| 6d 61 79 20 77 61 6e 74 | 20 74 6f 20 72 65 76 69 |may want| to revi|
|00005690| 65 77 20 10 31 2d 36 2d | 32 0e 65 31 2d 36 2d 34 |ew .1-6-|2.e1-6-4|
|000056a0| 0e 47 75 69 64 65 64 20 | 45 78 61 6d 70 6c 65 20 |.Guided |Example |
|000056b0| 34 2e 0f 0d 0a 00 53 65 | 63 74 69 6f 6e 20 31 2e |4.....Se|ction 1.|
|000056c0| 36 20 20 4f 74 68 65 72 | 20 54 79 70 65 73 20 6f |6  Other| Types o|
|000056d0| 66 20 45 71 75 61 74 69 | 6f 6e 73 20 20 20 20 20 |f Equati|ons     |
|000056e0| 20 20 20 20 20 20 20 20 | 11 34 44 32 32 32 32 32 |        |.4D22222|
|000056f0| 32 0d 0b 00 20 11 31 36 | 2e 20 20 53 6f 6c 76 65 |2... .16|.  Solve|
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|00005720| 00 20 20 20 20 20 20 20 | 20 20 32 20 20 20 20 20 |.       |  2     |
|00005730| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00005750| 20 20 20 20 20 0e 32 2d | 36 2d 32 39 61 0e 28 61 |     .2-|6-29a.(a|
|00005760| 29 0f 20 11 34 32 20 20 | 20 20 20 20 20 20 20 20 |). .42  |        |
|00005770| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005780| 20 20 11 31 0e 32 2d 36 | 2d 32 39 62 0e 28 62 29 |  .1.2-6|-29b.(b)|
|00005790| 0f 20 11 34 32 32 0d 0b | 00 20 20 20 20 20 20 20 |. .422..|.       |
|000057a0| 20 20 11 31 33 20 20 20 | 20 20 20 20 20 20 20 20 |  .13   |        |
|000057b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000057c0| 20 20 20 20 20 20 33 0d | 0a 00 0d 0b 00 20 20 20 |      3.|.....   |
|000057d0| 20 20 0e 32 2d 36 2d 32 | 39 63 0e 28 63 29 0f 20 |  .2-6-2|9c.(c). |
|000057e0| 31 32 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |12      |        |
|000057f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 0e 32 2d |        |     .2-|
|00005800| 36 2d 32 39 64 0e 28 64 | 29 0f 20 4e 6f 20 73 6f |6-29d.(d|). No so|
|00005810| 6c 75 74 69 6f 6e 0d 0a | 00 0d 0a 00 59 6f 75 20 |lution..|....You |
|00005820| 6d 61 79 20 77 61 6e 74 | 20 74 6f 20 72 65 76 69 |may want| to revi|
|00005830| 65 77 20 10 31 2d 36 2d | 32 0e 65 31 2d 36 2d 35 |ew .1-6-|2.e1-6-5|
|00005840| 0e 47 75 69 64 65 64 20 | 45 78 61 6d 70 6c 65 20 |.Guided |Example |
|00005850| 35 2e 0f 0d 0a 00 53 65 | 63 74 69 6f 6e 20 31 2e |5.....Se|ction 1.|
|00005860| 36 20 20 4f 74 68 65 72 | 20 54 79 70 65 73 20 6f |6  Other| Types o|
|00005870| 66 20 45 71 75 61 74 69 | 6f 6e 73 20 20 20 20 20 |f Equati|ons     |
|00005880| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005890| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000058a0| 20 20 11 32 33 2f 32 0d | 0b 00 20 11 31 37 2e 20 |  .23/2.|.. .17. |
|000058b0| 20 46 69 6e 64 20 74 68 | 65 20 72 65 61 6c 20 73 | Find th|e real s|
|000058c0| 6f 6c 75 74 69 6f 6e 73 | 20 6f 66 20 28 11 33 78 |olutions| of (.3x|
|000058d0| 20 11 31 2b 20 33 29 20 | 20 20 20 3d 20 38 2e 0d | .1+ 3) |   = 8..|
|000058e0| 0a 00 0d 0a 00 20 20 20 | 20 20 0e 61 32 2d 36 2d |.....   |  .a2-6-|
|000058f0| 34 37 61 0e 28 61 29 0f | 20 2d 31 20 20 20 20 20 |47a.(a).| -1     |
|00005900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005910| 20 20 20 20 20 20 0e 61 | 32 2d 36 2d 34 37 62 0e |      .a|2-6-47b.|
|00005920| 28 62 29 0f 20 31 0d 0a | 00 0d 0b 00 20 20 20 20 |(b). 1..|....    |
|00005930| 20 0e 61 32 2d 36 2d 34 | 37 63 0e 28 63 29 0f 20 | .a2-6-4|7c.(c). |
|00005940| 35 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |5       |        |
|00005950| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 0e 61 32 |        |     .a2|
|00005960| 2d 36 2d 34 37 64 0e 28 | 64 29 0f 20 4e 6f 20 72 |-6-47d.(|d). No r|
|00005970| 65 61 6c 20 73 6f 6c 75 | 74 69 6f 6e 0d 0a 00 0d |eal solu|tion....|
|00005980| 0a 00 59 6f 75 20 6d 61 | 79 20 77 61 6e 74 20 74 |..You ma|y want t|
|00005990| 6f 20 72 65 76 69 65 77 | 20 10 31 2d 36 2d 32 0e |o review| .1-6-2.|
|000059a0| 65 31 2d 36 2d 36 0e 47 | 75 69 64 65 64 20 45 78 |e1-6-6.G|uided Ex|
|000059b0| 61 6d 70 6c 65 20 36 2e | 0f 0d 0a 00 53 65 63 74 |ample 6.|....Sect|
|000059c0| 69 6f 6e 20 31 2e 36 20 | 20 4f 74 68 65 72 20 54 |ion 1.6 | Other T|
|000059d0| 79 70 65 73 20 6f 66 20 | 45 71 75 61 74 69 6f 6e |ypes of |Equation|
|000059e0| 73 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |s...    |        |
|000059f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005a00| 20 20 20 20 20 20 20 32 | 20 20 20 20 20 20 20 20 |       2|        |
|00005a10| 20 33 20 20 20 20 20 20 | 20 20 31 0d 0b 00 20 38 | 3      |  1... 8|
|00005a20| 2e 20 20 46 69 6e 64 20 | 74 68 65 20 72 65 61 6c |.  Find |the real|
|00005a30| 20 73 6f 6c 75 74 69 6f | 6e 73 20 6f 66 20 11 34 | solutio|ns of .4|
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|00005a50| 32 32 32 32 32 32 20 11 | 31 3d 20 11 34 32 32 20 |222222 .|1= .422 |
|00005a60| 11 31 2e 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |.1....  |        |
|00005a70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005a80| 20 20 20 20 20 20 36 11 | 33 78 20 11 31 2b 20 35 |      6.|3x .1+ 5|
|00005a90| 20 20 20 34 28 36 11 33 | 78 20 11 31 2b 20 35 29 |   4(6.3|x .1+ 5)|
|00005aa0| 20 20 20 32 38 0d 0a 00 | 0d 0a 00 20 20 20 20 20 |   28...|...     |
|00005ab0| 0e 61 32 2d 36 2d 35 37 | 61 0e 28 61 29 0f 20 2d |.a2-6-57|a.(a). -|
|00005ac0| 31 31 2f 32 20 20 20 20 | 20 20 20 20 20 20 20 20 |11/2    |        |
|00005ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 0e 61 32 2d |        |    .a2-|
|00005ae0| 36 2d 35 37 62 0e 28 62 | 29 0f 20 35 0d 0a 00 0d |6-57b.(b|). 5....|
|00005af0| 0b 00 20 20 20 20 20 0e | 61 32 2d 36 2d 35 37 63 |..     .|a2-6-57c|
|00005b00| 0e 28 63 29 0f 20 33 30 | 20 20 20 20 20 20 20 20 |.(c). 30|        |
|00005b10| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00005b30| 0f 20 4e 6f 20 72 65 61 | 6c 20 73 6f 6c 75 74 69 |. No rea|l soluti|
|00005b40| 6f 6e 0d 0a 00 0d 0a 00 | 59 6f 75 20 6d 61 79 20 |on......|You may |
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|00005b60| 31 2d 36 2d 32 0e 65 31 | 2d 36 2d 37 0e 47 75 69 |1-6-2.e1|-6-7.Gui|
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|00005b80| 0a 00 53 65 63 74 69 6f | 6e 20 31 2e 36 20 20 4f |..Sectio|n 1.6  O|
|00005b90| 74 68 65 72 20 54 79 70 | 65 73 20 6f 66 20 45 71 |ther Typ|es of Eq|
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|00005bb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005bc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 34 20 20 20 |        |    4   |
|00005bd0| 20 20 20 20 31 0d 0b 00 | 20 39 2e 20 20 46 69 6e |    1...| 9.  Fin|
|00005be0| 64 20 74 68 65 20 72 65 | 61 6c 20 73 6f 6c 75 74 |d the re|al solut|
|00005bf0| 69 6f 6e 73 20 6f 66 20 | 11 34 32 32 32 32 32 20 |ions of |.422222 |
|00005c00| 11 31 2d 20 11 34 32 32 | 32 32 32 20 11 31 3d 20 |.1- .422|222 .1= |
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|00005c30| 20 20 20 20 20 11 33 78 | 20 11 31 2b 20 33 20 20 |     .3x| .1+ 3  |
|00005c40| 20 11 33 78 20 11 31 2b | 20 32 0d 0a 00 0d 0a 00 | .3x .1+| 2......|
|00005c50| 20 20 20 20 20 0e 61 32 | 2d 36 2d 36 31 61 0e 28 |     .a2|-6-61a.(|
|00005c60| 61 29 0f 20 2d 31 20 20 | 20 20 20 20 20 20 20 20 |a). -1  |        |
|00005c70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00005c80| 20 0e 61 32 2d 36 2d 36 | 31 62 0e 28 62 29 0f 20 | .a2-6-6|1b.(b). |
|00005c90| 11 34 2b 11 31 31 0d 0a | 00 0d 0b 00 20 20 20 20 |.4+.11..|....    |
|00005ca0| 20 0e 61 32 2d 36 2d 36 | 31 63 0e 28 63 29 0f 20 | .a2-6-6|1c.(c). |
|00005cb0| 31 2c 20 30 20 20 20 20 | 20 20 20 20 20 20 20 20 |1, 0    |        |
|00005cc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 0e 61 32 |        |     .a2|
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|00005ce0| 65 61 6c 20 73 6f 6c 75 | 74 69 6f 6e 0d 0a 00 0d |eal solu|tion....|
|00005cf0| 0a 00 59 6f 75 20 6d 61 | 79 20 77 61 6e 74 20 74 |..You ma|y want t|
|00005d00| 6f 20 72 65 76 69 65 77 | 20 10 31 2d 36 2d 32 0e |o review| .1-6-2.|
|00005d10| 65 31 2d 36 2d 37 0e 47 | 75 69 64 65 64 20 45 78 |e1-6-7.G|uided Ex|
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+--------+-------------------------+-------------------------+--------+--------+