Interactive Algebra & Trigonometry - A Self Guided Study Companion

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Confidence	Program	Detection	Match Type	Support
`1%`	dexvert	Eclipse Tutorial `(other/eclipseTutorial)`	ext	Unsupported
`1%`	dexvert	JuggleKrazy Tutorial `(other/juggleKrazyTutorial)`	ext	Unsupported
`100%`	file	data	default
`100%`	gt2	Kopftext: 'TUTOR 06'	default (weak)
hex view
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|00000de0| 74 69 6f 6e 73 3a 20 32 | 2e 35 20 61 6e 64 20 33 |tions: 2|.5 and 3|
|00000df0| 2e 35 2e 20 20 54 72 79 | 20 63 68 65 63 6b 69 6e |.5.  Try| checkin|
|00000e00| 67 0d 0a 00 65 61 63 68 | 20 73 6f 6c 75 74 69 6f |g...each| solutio|
|00000e10| 6e 20 69 6e 20 74 68 65 | 20 6f 72 69 67 69 6e 61 |n in the| origina|
|00000e20| 6c 20 65 71 75 61 74 69 | 6f 6e 2e 0d 0a 00 00 17 |l equati|on......|
|00000e30| 4f 0f 0d 0b 00 12 30 57 | 72 6f 6e 67 2e 20 20 57 |O.....0W|rong.  W|
|00000e40| 65 20 73 6f 6c 76 65 20 | 74 68 65 20 67 69 76 65 |e solve |the give|
|00000e50| 6e 20 65 71 75 61 74 69 | 6f 6e 20 61 73 20 66 6f |n equati|on as fo|
|00000e60| 6c 6c 6f 77 73 2e 12 30 | 0d 0a 00 20 20 20 20 20 |llows..0|...     |
|00000e70| 20 11 32 32 0d 0b 00 20 | 20 20 20 20 11 33 78 20 | .22... |    .3x |
|00000e80| 20 11 31 2d 20 33 11 33 | 78 20 11 31 2d 20 31 38 | .1- 3.3|x .1- 18|
|00000e90| 20 3d 20 30 20 20 20 20 | 20 20 20 20 20 20 20 20 | = 0    |        |
|00000ea0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 47 69 76 |        | .2.1Giv|
|00000eb0| 65 6e 20 65 71 75 61 74 | 69 6f 6e 12 30 0d 0a 00 |en equat|ion.0...|
|00000ec0| 20 20 20 20 20 20 20 20 | 20 20 20 32 0d 0b 00 20 |        |   2... |
|00000ed0| 20 20 20 20 20 20 20 20 | 20 11 33 78 20 20 11 31 |        | .3x  .1|
|00000ee0| 2d 20 33 11 33 78 20 11 | 31 3d 20 31 38 20 20 20 |- 3.3x .|1= 18   |
|00000ef0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00000f00| 20 11 32 12 31 41 64 64 | 20 31 38 20 74 6f 20 62 | .2.1Add| 18 to b|
|00000f10| 6f 74 68 20 73 69 64 65 | 73 12 30 0d 0a 00 20 20 |oth side|s.0...  |
|00000f20| 20 32 20 20 20 20 20 20 | 20 20 20 20 11 31 33 20 | 2      |    .13 |
|00000f30| 11 32 32 20 20 20 20 20 | 20 20 20 20 20 11 31 33 |.22     |     .13|
|00000f40| 20 11 32 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | .22    |        |
|00000f50| 12 31 20 20 20 20 20 20 | 33 20 32 20 12 30 0d 0b |.1      |3 2 .0..|
|00000f60| 00 20 20 11 33 78 20 20 | 11 31 2d 20 33 11 33 78 |.  .3x  |.1- 3.3x|
|00000f70| 20 11 31 2b 20 28 2d 11 | 34 32 11 31 29 20 20 3d | .1+ (-.|42.1)  =|
|00000f80| 20 31 38 20 2b 20 28 2d | 11 34 32 11 31 29 20 20 | 18 + (-|.42.1)  |
|00000f90| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 41 |        |   .2.1A|
|00000fa0| 64 64 20 28 2d 2d 29 20 | 20 74 6f 20 62 6f 74 68 |dd (--) | to both|
|00000fb0| 20 73 69 64 65 73 12 30 | 0d 0b 00 20 20 20 20 20 | sides.0|...     |
|00000fc0| 20 20 20 20 20 20 20 20 | 20 11 31 32 20 20 20 20 |        | .12    |
|00000fd0| 20 20 20 20 20 20 20 20 | 32 20 20 20 20 20 20 20 |        |2       |
|00000fe0| 20 20 20 20 20 20 20 11 | 32 12 31 20 20 20 20 20 |       .|2.1     |
|00000ff0| 20 32 20 20 20 12 30 0d | 0a 00 20 20 20 20 20 20 | 2   .0.|..      |
|00001000| 20 20 20 20 20 20 20 20 | 11 31 33 20 11 32 32 20 |        |.13 .22 |
|00001010| 20 20 11 31 38 31 0d 0b | 00 20 20 20 20 20 20 20 |  .181..|.       |
|00001020| 20 20 28 11 33 78 20 11 | 31 2d 20 11 34 32 11 31 |  (.3x .|1- .42.1|
|00001030| 29 20 20 3d 20 11 34 32 | 32 20 20 20 20 20 20 20 |)  = .42|2       |
|00001040| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00001050| 31 50 65 72 66 65 63 74 | 20 73 71 75 61 72 65 20 |1Perfect| square |
|00001060| 74 72 69 6e 6f 6d 69 61 | 6c 12 30 0d 0b 00 20 20 |trinomia|l.0...  |
|00001070| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 31 32 20 |        |    .12 |
|00001080| 20 20 20 20 20 34 0d 0a | 00 20 20 20 20 20 20 20 |     4..|.       |
|00001090| 20 20 20 20 20 20 20 20 | 20 33 20 20 20 20 39 0d |        | 3    9.|
|000010a0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |..      |      .3|
|000010b0| 78 20 11 31 2d 20 11 34 | 32 20 11 31 3d 20 11 34 |x .1- .4|2 .1= .4|
|000010c0| 2b 32 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |+2      |        |
|000010d0| 20 20 20 20 20 20 11 32 | 12 31 45 78 74 72 61 63 |      .2|.1Extrac|
|000010e0| 74 20 73 71 75 61 72 65 | 20 72 6f 6f 74 73 12 30 |t square| roots.0|
|000010f0| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00001100| 20 20 20 11 31 32 20 20 | 20 20 32 0d 0a 00 20 20 |   .12  |  2...  |
|00001110| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001120| 20 20 33 20 20 20 39 20 | 20 20 20 33 20 20 20 39 |  3   9 |   3   9|
|00001130| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00001140| 20 20 20 11 33 78 20 11 | 31 3d 20 11 34 32 20 11 |   .3x .|1= .42 .|
|00001150| 31 2b 20 11 34 32 20 11 | 31 6f 72 20 11 34 32 20 |1+ .42 .|1or .42 |
|00001160| 11 31 2d 20 11 34 32 20 | 20 20 20 20 20 20 20 11 |.1- .42 |       .|
|00001170| 32 12 31 53 6f 6c 76 65 | 20 66 6f 72 20 78 12 30 |2.1Solve| for x.0|
|00001180| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00001190| 20 20 20 20 20 20 20 11 | 31 32 20 20 20 32 20 20 |       .|12   2  |
|000011a0| 20 20 32 20 20 20 32 0d | 0a 00 20 20 20 20 20 20 |  2   2.|..      |
|000011b0| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 31 |        |  .3x .1|
|000011c0| 3d 20 36 20 6f 72 20 2d | 33 20 20 20 20 20 20 20 |= 6 or -|3       |
|000011d0| 20 20 20 20 20 20 20 20 | 11 32 12 31 53 6f 6c 75 |        |.2.1Solu|
|000011e0| 74 69 6f 6e 73 12 30 0d | 0a 00 00 17 4f 0f 0d 0b |tions.0.|....O...|
|000011f0| 00 12 30 57 72 6f 6e 67 | 2e 20 20 57 65 20 73 6f |..0Wrong|.  We so|
|00001200| 6c 76 65 20 74 68 65 20 | 67 69 76 65 6e 20 65 71 |lve the |given eq|
|00001210| 75 61 74 69 6f 6e 20 61 | 73 20 66 6f 6c 6c 6f 77 |uation a|s follow|
|00001220| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 11 32 32 |s..0... |     .22|
|00001230| 0d 0b 00 20 20 20 20 20 | 11 33 78 20 20 11 31 2d |...     |.3x  .1-|
|00001240| 20 33 11 33 78 20 11 31 | 2d 20 31 38 20 3d 20 30 | 3.3x .1|- 18 = 0|
|00001250| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001260| 20 20 20 20 20 11 32 12 | 31 47 69 76 65 6e 20 65 |     .2.|1Given e|
|00001270| 71 75 61 74 69 6f 6e 12 | 30 0d 0a 00 20 20 20 20 |quation.|0...    |
|00001280| 20 20 20 20 20 20 20 32 | 0d 0b 00 20 20 20 20 20 |       2|...     |
|00001290| 20 20 20 20 20 11 33 78 | 20 20 11 31 2d 20 33 11 |     .3x|  .1- 3.|
|000012a0| 33 78 20 11 31 3d 20 31 | 38 20 20 20 20 20 20 20 |3x .1= 1|8       |
|000012b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|000012c0| 31 41 64 64 20 31 38 20 | 74 6f 20 62 6f 74 68 20 |1Add 18 |to both |
|000012d0| 73 69 64 65 73 12 30 0d | 0a 00 20 20 20 32 20 20 |sides.0.|..   2  |
|000012e0| 20 20 20 20 20 20 20 20 | 11 31 33 20 11 32 32 20 |        |.13 .22 |
|000012f0| 20 20 20 20 20 20 20 20 | 20 11 31 33 20 11 32 32 |        | .13 .22|
|00001300| 20 20 20 20 20 20 20 20 | 20 20 20 20 12 31 20 20 |        |    .1  |
|00001310| 20 20 20 20 33 20 32 20 | 12 30 0d 0b 00 20 20 11 |    3 2 |.0...  .|
|00001320| 33 78 20 20 11 31 2d 20 | 33 11 33 78 20 11 31 2b |3x  .1- |3.3x .1+|
|00001330| 20 28 2d 11 34 32 11 31 | 29 20 20 3d 20 31 38 20 | (-.42.1|)  = 18 |
|00001340| 2b 20 28 2d 11 34 32 11 | 31 29 20 20 20 20 20 20 |+ (-.42.|1)      |
|00001350| 20 20 20 20 20 20 20 11 | 32 12 31 41 64 64 20 28 |       .|2.1Add (|
|00001360| 2d 2d 29 20 20 74 6f 20 | 62 6f 74 68 20 73 69 64 |--)  to |both sid|
|00001370| 65 73 12 30 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |es.0... |        |
|00001380| 20 20 20 20 20 11 31 32 | 20 20 20 20 20 20 20 20 |     .12|        |
|00001390| 20 20 20 20 32 20 20 20 | 20 20 20 20 20 20 20 20 |    2   |        |
|000013a0| 20 20 20 11 32 12 31 20 | 20 20 20 20 20 32 20 20 |   .2.1 |     2  |
|000013b0| 20 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 | .0...  |        |
|000013c0| 20 20 20 20 11 31 33 20 | 11 32 32 20 20 20 11 31 |    .13 |.22   .1|
|000013d0| 38 31 0d 0b 00 20 20 20 | 20 20 20 20 20 20 28 11 |81...   |      (.|
|000013e0| 33 78 20 11 31 2d 20 11 | 34 32 11 31 29 20 20 3d |3x .1- .|42.1)  =|
|000013f0| 20 11 34 32 32 20 20 20 | 20 20 20 20 20 20 20 20 | .422   |        |
|00001400| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 50 65 72 |        | .2.1Per|
|00001410| 66 65 63 74 20 73 71 75 | 61 72 65 20 74 72 69 6e |fect squ|are trin|
|00001420| 6f 6d 69 61 6c 12 30 0d | 0b 00 20 20 20 20 20 20 |omial.0.|..      |
|00001430| 20 20 20 20 20 20 20 20 | 11 31 32 20 20 20 20 20 |        |.12     |
|00001440| 20 34 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 | 4...   |        |
|00001450| 20 20 20 20 20 33 20 20 | 20 20 39 0d 0b 00 20 20 |     3  |  9...  |
|00001460| 20 20 20 20 20 20 20 20 | 20 20 11 33 78 20 11 31 |        |  .3x .1|
|00001470| 2d 20 11 34 32 20 11 31 | 3d 20 11 34 2b 32 20 20 |- .42 .1|= .4+2  |
|00001480| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001490| 20 20 11 32 12 31 45 78 | 74 72 61 63 74 20 73 71 |  .2.1Ex|tract sq|
|000014a0| 75 61 72 65 20 72 6f 6f | 74 73 12 30 0d 0b 00 20 |uare roo|ts.0... |
|000014b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000014c0| 31 32 20 20 20 20 32 0d | 0a 00 20 20 20 20 20 20 |12    2.|..      |
|000014d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 33 20 |        |      3 |
|000014e0| 20 20 39 20 20 20 20 33 | 20 20 20 39 0d 0b 00 20 |  9    3|   9... |
|000014f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001500| 33 78 20 11 31 3d 20 11 | 34 32 20 11 31 2b 20 11 |3x .1= .|42 .1+ .|
|00001510| 34 32 20 11 31 6f 72 20 | 11 34 32 20 11 31 2d 20 |42 .1or |.42 .1- |
|00001520| 11 34 32 20 20 20 20 20 | 20 20 20 11 32 12 31 53 |.42     |   .2.1S|
|00001530| 6f 6c 76 65 20 66 6f 72 | 20 78 12 30 0d 0b 00 20 |olve for| x.0... |
|00001540| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001550| 20 20 20 11 31 32 20 20 | 20 32 20 20 20 20 32 20 |   .12  | 2    2 |
|00001560| 20 20 32 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |  2...  |        |
|00001570| 20 20 20 20 20 20 11 33 | 78 20 11 31 3d 20 36 20 |      .3|x .1= 6 |
|00001580| 6f 72 20 2d 33 20 20 20 | 20 20 20 20 20 20 20 20 |or -3   |        |
|00001590| 20 20 20 20 11 32 12 31 | 53 6f 6c 75 74 69 6f 6e |    .2.1|Solution|
|000015a0| 73 12 30 0d 0a 00 00 17 | 4f 0f 0d 0b 00 12 30 57 |s.0.....|O.....0W|
|000015b0| 72 6f 6e 67 2e 20 20 57 | 65 20 73 6f 6c 76 65 20 |rong.  W|e solve |
|000015c0| 74 68 65 20 67 69 76 65 | 6e 20 65 71 75 61 74 69 |the give|n equati|
|000015d0| 6f 6e 20 61 73 20 66 6f | 6c 6c 6f 77 73 2e 12 30 |on as fo|llows..0|
|000015e0| 0d 0a 00 20 20 20 20 20 | 20 11 32 32 0d 0b 00 20 |...     | .22... |
|000015f0| 20 20 20 20 11 33 78 20 | 20 11 31 2d 20 33 11 33 |    .3x | .1- 3.3|
|00001600| 78 20 11 31 2d 20 31 38 | 20 3d 20 30 20 20 20 20 |x .1- 18| = 0    |
|00001610| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001620| 20 11 32 12 31 47 69 76 | 65 6e 20 65 71 75 61 74 | .2.1Giv|en equat|
|00001630| 69 6f 6e 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |ion.0...|        |
|00001640| 20 20 20 32 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |   2... |        |
|00001650| 20 11 33 78 20 20 11 31 | 2d 20 33 11 33 78 20 11 | .3x  .1|- 3.3x .|
|00001660| 31 3d 20 31 38 20 20 20 | 20 20 20 20 20 20 20 20 |1= 18   |        |
|00001670| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 41 64 64 |        | .2.1Add|
|00001680| 20 31 38 20 74 6f 20 62 | 6f 74 68 20 73 69 64 65 | 18 to b|oth side|
|00001690| 73 12 30 0d 0a 00 20 20 | 20 32 20 20 20 20 20 20 |s.0...  | 2      |
|000016a0| 20 20 20 20 11 31 33 20 | 11 32 32 20 20 20 20 20 |    .13 |.22     |
|000016b0| 20 20 20 20 20 11 31 33 | 20 11 32 32 20 20 20 20 |     .13| .22    |
|000016c0| 20 20 20 20 20 20 20 20 | 12 31 20 20 20 20 20 20 |        |.1      |
|000016d0| 33 20 32 20 12 30 0d 0b | 00 20 20 11 33 78 20 20 |3 2 .0..|.  .3x  |
|000016e0| 11 31 2d 20 33 11 33 78 | 20 11 31 2b 20 28 2d 11 |.1- 3.3x| .1+ (-.|
|000016f0| 34 32 11 31 29 20 20 3d | 20 31 38 20 2b 20 28 2d |42.1)  =| 18 + (-|
|00001700| 11 34 32 11 31 29 20 20 | 20 20 20 20 20 20 20 20 |.42.1)  |        |
|00001710| 20 20 20 11 32 12 31 41 | 64 64 20 28 2d 2d 29 20 |   .2.1A|dd (--) |
|00001720| 20 74 6f 20 62 6f 74 68 | 20 73 69 64 65 73 12 30 | to both| sides.0|
|00001730| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00001740| 20 11 31 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | .12    |        |
|00001750| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |2       |       .|
|00001760| 32 12 31 20 20 20 20 20 | 20 32 20 20 20 12 30 0d |2.1     | 2   .0.|
|00001770| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00001780| 11 31 33 20 11 32 32 20 | 20 20 11 31 38 31 0d 0b |.13 .22 |  .181..|
|00001790| 00 20 20 20 20 20 20 20 | 20 20 28 11 33 78 20 11 |.       |  (.3x .|
|000017a0| 31 2d 20 11 34 32 11 31 | 29 20 20 3d 20 11 34 32 |1- .42.1|)  = .42|
|000017b0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|000017c0| 20 20 20 20 20 11 32 12 | 31 50 65 72 66 65 63 74 |     .2.|1Perfect|
|000017d0| 20 73 71 75 61 72 65 20 | 74 72 69 6e 6f 6d 69 61 | square |trinomia|
|000017e0| 6c 12 30 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |l.0...  |        |
|000017f0| 20 20 20 20 11 31 32 20 | 20 20 20 20 20 34 0d 0a |    .12 |     4..|
|00001800| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001810| 20 33 20 20 20 20 39 0d | 0b 00 20 20 20 20 20 20 | 3    9.|..      |
|00001820| 20 20 20 20 20 20 11 33 | 78 20 11 31 2d 20 11 34 |      .3|x .1- .4|
|00001830| 32 20 11 31 3d 20 11 34 | 2b 32 20 20 20 20 20 20 |2 .1= .4|+2      |
|00001840| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00001850| 12 31 45 78 74 72 61 63 | 74 20 73 71 75 61 72 65 |.1Extrac|t square|
|00001860| 20 72 6f 6f 74 73 12 30 | 0d 0b 00 20 20 20 20 20 | roots.0|...     |
|00001870| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 32 20 20 |        |   .12  |
|00001880| 20 20 32 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |  2...  |        |
|00001890| 20 20 20 20 20 20 20 20 | 20 20 33 20 20 20 39 20 |        |  3   9 |
|000018a0| 20 20 20 33 20 20 20 39 | 0d 0b 00 20 20 20 20 20 |   3   9|...     |
|000018b0| 20 20 20 20 20 20 20 20 | 20 20 20 11 33 78 20 11 |        |   .3x .|
|000018c0| 31 3d 20 11 34 32 20 11 | 31 2b 20 11 34 32 20 11 |1= .42 .|1+ .42 .|
|000018d0| 31 6f 72 20 11 34 32 20 | 11 31 2d 20 11 34 32 20 |1or .42 |.1- .42 |
|000018e0| 20 20 20 20 20 20 20 11 | 32 12 31 53 6f 6c 76 65 |       .|2.1Solve|
|000018f0| 20 66 6f 72 20 78 12 30 | 0d 0b 00 20 20 20 20 20 | for x.0|...     |
|00001900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001910| 31 32 20 20 20 32 20 20 | 20 20 32 20 20 20 32 0d |12   2  |  2   2.|
|00001920| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00001930| 20 20 11 33 78 20 11 31 | 3d 20 36 20 6f 72 20 2d |  .3x .1|= 6 or -|
|00001940| 33 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |3       |        |
|00001950| 11 32 12 31 53 6f 6c 75 | 74 69 6f 6e 73 12 30 0d |.2.1Solu|tions.0.|
|00001960| 0a 00 00 17 4f 03 0d 0b | 00 52 69 67 68 74 2e 20 |....O...|.Right. |
|00001970| 20 54 68 65 20 67 69 76 | 65 6e 20 65 71 75 61 74 | The giv|en equat|
|00001980| 69 6f 6e 20 68 61 73 20 | 74 77 6f 20 73 6f 6c 75 |ion has |two solu|
|00001990| 74 69 6f 6e 73 3a 20 2d | 33 20 61 6e 64 20 36 2e |tions: -|3 and 6.|
|000019a0| 0d 0a 00 00 17 4f 0d 0d | 0b 00 4e 6f 2e 20 20 42 |.....O..|..No.  B|
|000019b0| 65 67 69 6e 20 62 79 20 | 73 75 62 74 72 61 63 74 |egin by |subtract|
|000019c0| 69 6e 67 20 33 20 66 72 | 6f 6d 20 62 6f 74 68 20 |ing 3 fr|om both |
|000019d0| 73 69 64 65 73 20 61 6e | 64 20 64 69 76 69 64 69 |sides an|d dividi|
|000019e0| 6e 67 20 62 6f 74 68 20 | 73 69 64 65 73 20 62 79 |ng both |sides by|
|000019f0| 20 32 2e 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 | 2..0...|        |
|00001a00| 11 32 32 20 20 20 20 20 | 20 20 20 20 11 31 33 0d |.22     |    .13.|
|00001a10| 0b 00 20 20 20 20 20 20 | 20 11 33 78 20 20 11 31 |..      | .3x  .1|
|00001a20| 2b 20 34 11 33 78 20 11 | 31 3d 20 2d 11 34 32 0d |+ 4.3x .|1= -.42.|
|00001a30| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00001a40| 20 20 20 20 11 31 32 0d | 0a 00 20 20 20 11 32 32 |    .12.|..   .22|
|00001a50| 20 20 20 20 20 20 20 20 | 20 32 20 20 20 20 11 31 |        | 2    .1|
|00001a60| 33 20 20 20 20 11 32 32 | 20 20 20 20 20 20 20 20 |3    .22|        |
|00001a70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001a80| 20 20 20 12 31 20 20 20 | 20 20 32 20 12 30 0d 0b |   .1   |  2 .0..|
|00001a90| 00 20 20 11 33 78 20 20 | 11 31 2b 20 34 11 33 78 |.  .3x  |.1+ 4.3x|
|00001aa0| 20 11 31 2b 20 32 20 20 | 3d 20 2d 11 34 32 20 11 | .1+ 2  |= -.42 .|
|00001ab0| 31 2b 20 32 20 20 20 20 | 20 20 20 20 20 20 20 20 |1+ 2    |        |
|00001ac0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ad0| 11 32 12 31 41 64 64 20 | 32 20 20 74 6f 20 62 6f |.2.1Add |2  to bo|
|00001ae0| 74 68 20 73 69 64 65 73 | 12 30 0d 0b 00 20 20 20 |th sides|.0...   |
|00001af0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00001b00| 31 32 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |12...   |        |
|00001b10| 20 20 11 32 32 20 20 20 | 11 31 35 0d 0b 00 20 20 |  .22   |.15...  |
|00001b20| 20 20 20 20 28 11 33 78 | 20 11 31 2b 20 32 29 20 |    (.3x| .1+ 2) |
|00001b30| 20 3d 20 11 34 32 20 20 | 20 20 20 20 20 20 20 20 | = .42  |        |
|00001b40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001b50| 20 20 20 20 20 20 20 11 | 32 12 31 50 65 72 66 65 |       .|2.1Perfe|
|00001b60| 63 74 20 73 71 75 61 72 | 65 20 74 72 69 6e 6f 6d |ct squar|e trinom|
|00001b70| 69 61 6c 12 30 0d 0b 00 | 20 20 20 20 20 20 20 20 |ial.0...|        |
|00001b80| 20 20 20 20 20 20 20 20 | 20 11 31 32 0d 0b 00 20 |        | .12... |
|00001b90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ba0| 20 20 20 20 11 34 67 32 | 20 20 20 20 20 20 20 67 |    .4g2|       g|
|00001bb0| 32 20 20 20 20 20 20 67 | 32 20 20 20 20 20 44 32 |2      g|2     D2|
|00001bc0| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |2...    |        |
|00001bd0| 20 20 20 20 20 20 20 20 | 66 20 11 31 35 20 20 20 |        |f .15   |
|00001be0| 20 20 20 11 34 66 20 11 | 31 35 20 20 20 20 20 11 |   .4f .|15     .|
|00001bf0| 34 66 20 11 31 32 20 20 | 20 20 11 34 53 20 11 31 |4f .12  |  .4S .1|
|00001c00| 31 30 0d 0b 00 20 20 20 | 20 20 20 20 20 20 11 33 |10...   |      .3|
|00001c10| 78 20 11 31 2b 20 32 20 | 3d 20 11 34 2b 20 66 20 |x .1+ 2 |= .4+ f |
|00001c20| 20 32 20 11 31 3d 20 11 | 34 2b 20 66 20 20 32 20 | 2 .1= .|4+ f  2 |
|00001c30| 2a 20 20 66 20 20 32 20 | 11 31 3d 20 11 34 2b 32 |*  f  2 |.1= .4+2|
|00001c40| 32 32 32 20 20 20 11 32 | 12 31 45 78 74 72 61 63 |222   .2|.1Extrac|
|00001c50| 74 20 73 71 75 61 72 65 | 20 72 6f 6f 74 73 12 30 |t square| roots.0|
|00001c60| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
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|00001c80| 20 20 11 34 76 20 20 20 | 11 31 32 20 20 20 11 34 |  .4v   |.12   .4|
|00001c90| 76 20 20 20 11 31 32 20 | 20 20 20 20 20 32 0d 0a |v   .12 |     2..|
|00001ca0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001cb0| 20 20 20 20 20 20 20 20 | 11 34 44 32 32 0d 0b 00 |        |.4D22...|
|00001cc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00001ce0| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 33 78 20 |        |    .3x |
|00001cf0| 11 31 3d 20 2d 32 20 11 | 34 2b 20 32 32 32 32 20 |.1= -2 .|4+ 2222 |
|00001d00| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001d10| 20 20 20 20 20 20 20 20 | 11 32 12 31 53 6f 6c 76 |        |.2.1Solv|
|00001d20| 65 20 66 6f 72 20 78 12 | 30 0d 0b 00 20 20 20 20 |e for x.|0...    |
|00001d30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001d40| 20 20 20 20 11 31 32 0d | 0a 00 00 17 4f 03 0d 0b |    .12.|....O...|
|00001d50| 00 52 69 67 68 74 2e 20 | 20 43 68 65 63 6b 20 79 |.Right. | Check y|
|00001d60| 6f 75 72 20 61 6e 73 77 | 65 72 20 69 6e 20 74 68 |our answ|er in th|
|00001d70| 65 20 6f 72 69 67 69 6e | 61 6c 20 73 74 61 74 65 |e origin|al state|
|00001d80| 6d 65 6e 74 20 6f 66 20 | 74 68 65 20 70 72 6f 62 |ment of |the prob|
|00001d90| 6c 65 6d 2e 0d 0a 00 00 | 17 4f 0d 0d 0b 00 4e 6f |lem.....|.O....No|
|00001da0| 2e 20 20 42 65 67 69 6e | 20 62 79 20 73 75 62 74 |.  Begin| by subt|
|00001db0| 72 61 63 74 69 6e 67 20 | 33 20 66 72 6f 6d 20 62 |racting |3 from b|
|00001dc0| 6f 74 68 20 73 69 64 65 | 73 20 61 6e 64 20 64 69 |oth side|s and di|
|00001dd0| 76 69 64 69 6e 67 20 62 | 6f 74 68 20 73 69 64 65 |viding b|oth side|
|00001de0| 73 20 62 79 20 32 2e 12 | 30 0d 0a 00 20 20 20 20 |s by 2..|0...    |
|00001df0| 20 20 20 20 11 32 32 20 | 20 20 20 20 20 20 20 20 |    .22 |        |
|00001e00| 11 31 33 0d 0b 00 20 20 | 20 20 20 20 20 11 33 78 |.13...  |     .3x|
|00001e10| 20 20 11 31 2b 20 34 11 | 33 78 20 11 31 3d 20 2d |  .1+ 4.|3x .1= -|
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|00001e70| 20 20 20 20 20 20 20 12 | 31 20 20 20 20 20 32 20 |       .|1     2 |
|00001e80| 12 30 0d 0b 00 20 20 11 | 33 78 20 20 11 31 2b 20 |.0...  .|3x  .1+ |
|00001e90| 34 11 33 78 20 11 31 2b | 20 32 20 20 3d 20 2d 11 |4.3x .1+| 2  = -.|
|00001ea0| 34 32 20 11 31 2b 20 32 | 20 20 20 20 20 20 20 20 |42 .1+ 2|        |
|00001eb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001ec0| 20 20 20 20 11 32 12 31 | 41 64 64 20 32 20 20 74 |    .2.1|Add 2  t|
|00001ed0| 6f 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0b |o both s|ides.0..|
|00001ee0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00001ef0| 20 20 20 11 31 32 0d 0a | 00 20 20 20 20 20 20 20 |   .12..|.       |
|00001f00| 20 20 20 20 20 20 11 32 | 32 20 20 20 11 31 35 0d |      .2|2   .15.|
|00001f10| 0b 00 20 20 20 20 20 20 | 28 11 33 78 20 11 31 2b |..      |(.3x .1+|
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|00001f30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00001f40| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 50 |        |   .2.1P|
|00001f50| 65 72 66 65 63 74 20 73 | 71 75 61 72 65 20 74 72 |erfect s|quare tr|
|00001f60| 69 6e 6f 6d 69 61 6c 12 | 30 0d 0b 00 20 20 20 20 |inomial.|0...    |
|00001f70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 32 |        |     .12|
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|00001fe0| 20 20 20 11 34 66 20 11 | 31 32 20 20 20 20 11 34 |   .4f .|12    .4|
|00001ff0| 53 20 11 31 31 30 0d 0b | 00 20 20 20 20 20 20 20 |S .110..|.       |
|00002000| 20 20 11 33 78 20 11 31 | 2b 20 32 20 3d 20 11 34 |  .3x .1|+ 2 = .4|
|00002010| 2b 20 66 20 20 32 20 11 | 31 3d 20 11 34 2b 20 66 |+ f  2 .|1= .4+ f|
|00002020| 20 20 32 20 2a 20 20 66 | 20 20 32 20 11 31 3d 20 |  2 *  f|  2 .1= |
|00002030| 11 34 2b 32 32 32 32 20 | 20 20 11 32 12 31 45 78 |.4+2222 |  .2.1Ex|
|00002040| 74 72 61 63 74 20 73 71 | 75 61 72 65 20 72 6f 6f |tract sq|uare roo|
|00002050| 74 73 12 30 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |ts.0... |        |
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|000020e0| 11 33 78 20 11 31 3d 20 | 2d 32 20 11 34 2b 20 32 |.3x .1= |-2 .4+ 2|
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|00002100| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |        |    .2.1|
|00002110| 53 6f 6c 76 65 20 66 6f | 72 20 78 12 30 0d 0b 00 |Solve fo|r x.0...|
|00002120| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002130| 20 20 20 20 20 20 20 20 | 11 31 32 0d 0a 00 00 17 |        |.12.....|
|00002140| 4f 0d 0d 0b 00 4e 6f 2e | 20 20 42 65 67 69 6e 20 |O....No.|  Begin |
|00002150| 62 79 20 73 75 62 74 72 | 61 63 74 69 6e 67 20 33 |by subtr|acting 3|
|00002160| 20 66 72 6f 6d 20 62 6f | 74 68 20 73 69 64 65 73 | from bo|th sides|
|00002170| 20 61 6e 64 20 64 69 76 | 69 64 69 6e 67 20 62 6f | and div|iding bo|
|00002180| 74 68 20 73 69 64 65 73 | 20 62 79 20 32 2e 12 30 |th sides| by 2..0|
|00002190| 0d 0a 00 20 20 20 20 20 | 20 20 20 11 32 32 20 20 |...     |   .22  |
|000021a0| 20 20 20 20 20 20 20 11 | 31 33 0d 0b 00 20 20 20 |       .|13...   |
|000021b0| 20 20 20 20 11 33 78 20 | 20 11 31 2b 20 34 11 33 |    .3x | .1+ 4.3|
|000021c0| 78 20 11 31 3d 20 2d 11 | 34 32 0d 0b 00 20 20 20 |x .1= -.|42...   |
|000021d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000021e0| 31 32 0d 0a 00 20 20 20 | 11 32 32 20 20 20 20 20 |12...   |.22     |
|000021f0| 20 20 20 20 32 20 20 20 | 20 11 31 33 20 20 20 20 |    2   | .13    |
|00002200| 11 32 32 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.22     |        |
|00002210| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 12 31 |        |      .1|
|00002220| 20 20 20 20 20 32 20 12 | 30 0d 0b 00 20 20 11 33 |     2 .|0...  .3|
|00002230| 78 20 20 11 31 2b 20 34 | 11 33 78 20 11 31 2b 20 |x  .1+ 4|.3x .1+ |
|00002240| 32 20 20 3d 20 2d 11 34 | 32 20 11 31 2b 20 32 20 |2  = -.4|2 .1+ 2 |
|00002250| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002260| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 41 |        |   .2.1A|
|00002270| 64 64 20 32 20 20 74 6f | 20 62 6f 74 68 20 73 69 |dd 2  to| both si|
|00002280| 64 65 73 12 30 0d 0b 00 | 20 20 20 20 20 20 20 20 |des.0...|        |
|00002290| 20 20 20 20 20 20 20 20 | 20 20 11 31 32 0d 0a 00 |        |  .12...|
|000022a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 32 |        |     .22|
|000022b0| 20 20 20 11 31 35 0d 0b | 00 20 20 20 20 20 20 28 |   .15..|.      (|
|000022c0| 11 33 78 20 11 31 2b 20 | 32 29 20 20 3d 20 11 34 |.3x .1+ |2)  = .4|
|000022d0| 32 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |2       |        |
|000022e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000022f0| 20 20 11 32 12 31 50 65 | 72 66 65 63 74 20 73 71 |  .2.1Pe|rfect sq|
|00002300| 75 61 72 65 20 74 72 69 | 6e 6f 6d 69 61 6c 12 30 |uare tri|nomial.0|
|00002310| 0d 0b 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
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|00002330| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00002340| 34 67 32 20 20 20 20 20 | 20 20 67 32 20 20 20 20 |4g2     |  g2    |
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|00002380| 66 20 11 31 35 20 20 20 | 20 20 11 34 66 20 11 31 |f .15   |  .4f .1|
|00002390| 32 20 20 20 20 11 34 53 | 20 11 31 31 30 0d 0b 00 |2    .4S| .110...|
|000023a0| 20 20 20 20 20 20 20 20 | 20 11 33 78 20 11 31 2b |        | .3x .1+|
|000023b0| 20 32 20 3d 20 11 34 2b | 20 66 20 20 32 20 11 31 | 2 = .4+| f  2 .1|
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|000023d0| 20 32 20 11 31 3d 20 11 | 34 2b 32 32 32 32 20 20 | 2 .1= .|4+2222  |
|000023e0| 20 11 32 12 31 45 78 74 | 72 61 63 74 20 73 71 75 | .2.1Ext|ract squ|
|000023f0| 61 72 65 20 72 6f 6f 74 | 73 12 30 0d 0b 00 20 20 |are root|s.0...  |
|00002400| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002410| 11 34 76 20 20 20 11 31 | 32 20 20 20 20 11 34 76 |.4v   .1|2    .4v|
|00002420| 20 20 20 11 31 32 20 20 | 20 11 34 76 20 20 20 11 |   .12  | .4v   .|
|00002430| 31 32 20 20 20 20 20 20 | 32 0d 0a 00 20 20 20 20 |12      |2...    |
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|00002450| 20 20 20 11 34 44 32 32 | 0d 0b 00 20 20 20 20 20 |   .4D22|...     |
|00002460| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00002480| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 3d 20 2d |       .|3x .1= -|
|00002490| 32 20 11 34 2b 20 32 32 | 32 32 20 20 20 20 20 20 |2 .4+ 22|22      |
|000024a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000024b0| 20 20 20 11 32 12 31 53 | 6f 6c 76 65 20 66 6f 72 |   .2.1S|olve for|
|000024c0| 20 78 12 30 0d 0b 00 20 | 20 20 20 20 20 20 20 20 | x.0... |        |
|000024d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
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|000024f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002500| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002510| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002520| 11 32 32 0d 0b 00 11 31 | 57 72 6f 6e 67 2e 20 20 |.22....1|Wrong.  |
|00002530| 54 68 65 20 73 74 61 6e | 64 61 72 64 20 66 6f 72 |The stan|dard for|
|00002540| 6d 20 6f 66 20 61 20 71 | 75 61 64 72 61 74 69 63 |m of a q|uadratic|
|00002550| 20 65 71 75 61 74 69 6f | 6e 20 69 73 20 11 33 61 | equatio|n is .3a|
|00002560| 78 20 20 11 31 2b 20 11 | 33 62 78 20 11 31 2b 20 |x  .1+ .|3bx .1+ |
|00002570| 11 33 63 20 11 31 3d 20 | 30 2e 0d 0a 00 54 68 75 |.3c .1= |0....Thu|
|00002580| 73 2c 20 77 65 20 70 75 | 74 20 11 33 78 11 31 28 |s, we pu|t .3x.1(|
|00002590| 31 30 20 2d 20 11 33 78 | 11 31 29 20 3d 20 35 20 |10 - .3x|.1) = 5 |
|000025a0| 69 6e 20 73 74 61 6e 64 | 61 72 64 20 66 6f 72 6d |in stand|ard form|
|000025b0| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 12 30 0d 0a | as foll|ows..0..|
|000025c0| 00 0d 0b 00 20 20 20 20 | 20 20 11 33 78 11 31 28 |....    |  .3x.1(|
|000025d0| 31 30 20 2d 20 11 33 78 | 11 31 29 20 3d 20 35 20 |10 - .3x|.1) = 5 |
|000025e0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 47 69 76 |        | .2.1Giv|
|000025f0| 65 6e 20 65 71 75 61 74 | 69 6f 6e 12 30 0d 0a 00 |en equat|ion.0...|
|00002600| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 32 0d |        |      2.|
|00002610| 0b 00 20 20 20 20 20 20 | 20 11 31 31 30 11 33 78 |..      | .110.3x|
|00002620| 20 11 31 2d 20 11 33 78 | 20 20 11 31 3d 20 35 20 | .1- .3x|  .1= 5 |
|00002630| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 52 65 6d |        | .2.1Rem|
|00002640| 6f 76 65 20 70 61 72 65 | 6e 74 68 65 73 65 73 12 |ove pare|ntheses.|
|00002650| 30 0d 0a 00 20 20 20 20 | 32 0d 0b 00 20 20 11 31 |0...    |2...  .1|
|00002660| 2d 11 33 78 20 20 11 31 | 2b 20 31 30 11 33 78 20 |-.3x  .1|+ 10.3x |
|00002670| 11 31 2d 20 35 20 3d 20 | 30 20 20 20 20 20 20 20 |.1- 5 = |0       |
|00002680| 20 20 20 11 32 12 31 53 | 74 61 6e 64 61 72 64 20 |   .2.1S|tandard |
|00002690| 66 6f 72 6d 12 30 0d 0a | 00 00 17 4f 08 20 20 20 |form.0..|...O.   |
|000026a0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000026b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000026c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000026d0| 20 20 20 20 11 32 32 0d | 0b 00 11 31 57 72 6f 6e |    .22.|...1Wron|
|000026e0| 67 2e 20 20 54 68 65 20 | 73 74 61 6e 64 61 72 64 |g.  The |standard|
|000026f0| 20 66 6f 72 6d 20 6f 66 | 20 61 20 71 75 61 64 72 | form of| a quadr|
|00002700| 61 74 69 63 20 65 71 75 | 61 74 69 6f 6e 20 69 73 |atic equ|ation is|
|00002710| 20 11 33 61 78 20 20 11 | 31 2b 20 11 33 62 78 20 | .3ax  .|1+ .3bx |
|00002720| 11 31 2b 20 11 33 63 20 | 11 31 3d 20 30 2e 0d 0a |.1+ .3c |.1= 0...|
|00002730| 00 54 68 75 73 2c 20 77 | 65 20 70 75 74 20 11 33 |.Thus, w|e put .3|
|00002740| 78 11 31 28 31 30 20 2d | 20 11 33 78 11 31 29 20 |x.1(10 -| .3x.1) |
|00002750| 3d 20 35 20 69 6e 20 73 | 74 61 6e 64 61 72 64 20 |= 5 in s|tandard |
|00002760| 66 6f 72 6d 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |form as |follows.|
|00002770| 20 20 20 12 30 0d 0a 00 | 0d 0b 00 20 20 20 20 20 |   .0...|...     |
|00002780| 20 11 33 78 11 31 28 31 | 30 20 2d 20 11 33 78 11 | .3x.1(1|0 - .3x.|
|00002790| 31 29 20 3d 20 35 20 20 | 20 20 20 20 20 20 20 20 |1) = 5  |        |
|000027a0| 11 32 12 31 47 69 76 65 | 6e 20 65 71 75 61 74 69 |.2.1Give|n equati|
|000027b0| 6f 6e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |on.0... |        |
|000027c0| 20 20 20 20 20 32 0d 0b | 00 20 20 20 20 20 20 20 |     2..|.       |
|000027d0| 11 31 31 30 11 33 78 20 | 11 31 2d 20 11 33 78 20 |.110.3x |.1- .3x |
|000027e0| 20 11 31 3d 20 35 20 20 | 20 20 20 20 20 20 20 20 | .1= 5  |        |
|000027f0| 11 32 12 31 52 65 6d 6f | 76 65 20 70 61 72 65 6e |.2.1Remo|ve paren|
|00002800| 74 68 65 73 65 73 12 30 | 0d 0a 00 20 20 20 20 32 |theses.0|...    2|
|00002810| 0d 0b 00 20 20 11 31 2d | 11 33 78 20 20 11 31 2b |...  .1-|.3x  .1+|
|00002820| 20 31 30 11 33 78 20 11 | 31 2d 20 35 20 3d 20 30 | 10.3x .|1- 5 = 0|
|00002830| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 53 74 |        |  .2.1St|
|00002840| 61 6e 64 61 72 64 20 66 | 6f 72 6d 12 30 0d 0a 00 |andard f|orm.0...|
|00002850| 00 17 4f 04 0d 0b 00 52 | 69 67 68 74 2e 20 20 4e |..O....R|ight.  N|
|00002860| 6f 74 65 20 74 68 61 74 | 20 77 65 20 63 6f 75 6c |ote that| we coul|
|00002870| 64 20 61 6c 73 6f 20 77 | 72 69 74 65 20 11 33 78 |d also w|rite .3x|
|00002880| 11 31 28 31 30 20 2d 11 | 33 78 11 31 29 20 3d 20 |.1(10 -.|3x.1) = |
|00002890| 35 20 69 6e 20 73 74 61 | 6e 64 61 72 64 20 66 6f |5 in sta|ndard fo|
|000028a0| 72 6d 20 61 73 0d 0a 00 | 20 11 32 32 0d 0b 00 11 |rm as...| .22....|
|000028b0| 33 78 20 20 11 31 2d 20 | 31 30 11 33 78 20 11 31 |3x  .1- |10.3x .1|
|000028c0| 2b 20 35 20 3d 20 30 2e | 0d 0a 00 00 17 4f 03 0d |+ 5 = 0.|.....O..|
|000028d0| 0b 00 41 6c 6d 6f 73 74 | 2e 20 20 43 68 65 63 6b |..Almost|.  Check|
|000028e0| 20 79 6f 75 72 20 77 6f | 72 6b 2e 0d 0a 00 00 17 | your wo|rk......|
|000028f0| 4f 07 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |O.      |        |
|00002900| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002910| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00002920| 32 0d 0b 00 11 31 57 72 | 6f 6e 67 2e 20 20 54 68 |2....1Wr|ong.  Th|
|00002930| 65 20 65 71 75 61 74 69 | 6f 6e 20 69 6e 20 73 74 |e equati|on in st|
|00002940| 61 6e 64 61 72 64 20 66 | 6f 72 6d 20 69 73 20 31 |andard f|orm is 1|
|00002950| 38 11 33 78 20 20 11 31 | 2d 20 32 34 11 33 78 20 |8.3x  .1|- 24.3x |
|00002960| 11 31 2b 20 38 20 3d 20 | 30 2e 20 20 57 69 74 68 |.1+ 8 = |0.  With|
|00002970| 20 11 33 61 20 11 31 3d | 20 31 38 2c 20 0d 0a 00 | .3a .1=| 18, ...|
|00002980| 11 33 62 20 11 31 3d 20 | 2d 32 34 2c 20 61 6e 64 |.3b .1= |-24, and|
|00002990| 20 11 33 63 20 11 31 3d | 20 38 2c 20 77 65 20 73 | .3c .1=| 8, we s|
|000029a0| 6f 6c 76 65 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |olve as |follows.|
|000029b0| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|000029c0| 20 20 20 20 20 20 20 20 | 20 11 34 67 32 32 32 32 |        | .4g2222|
|000029d0| 32 32 32 32 32 32 32 32 | 32 32 32 32 0d 0b 00 20 |22222222|2222... |
|000029e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00002a00| 20 20 20 20 20 20 20 11 | 31 2d 28 2d 32 34 29 20 |       .|1-(-24) |
|00002a10| 11 34 2b 20 76 20 11 31 | 28 2d 32 34 29 20 20 2d |.4+ v .1|(-24)  -|
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|00002a40| 0b 00 20 20 20 20 20 11 | 33 78 20 11 31 3d 20 11 |..     .|3x .1= .|
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|00002a60| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 20 11 31 |22222222|22222 .1|
|00002a70| 3d 20 11 34 32 32 32 32 | 32 32 20 11 31 3d 20 11 |= .42222|22 .1= .|
|00002a80| 34 32 32 20 11 31 3d 20 | 11 34 32 0d 0b 00 20 20 |422 .1= |.42...  |
|00002a90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002aa0| 20 20 11 31 32 28 31 38 | 29 20 20 20 20 20 20 20 |  .12(18|)       |
|00002ab0| 20 20 20 20 20 20 20 20 | 20 20 33 36 20 20 20 20 |        |  36    |
|00002ac0| 20 33 36 20 20 20 33 0d | 0a 00 00 17 4f 07 20 20 | 36   3.|....O.  |
|00002ad0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ae0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002af0| 20 20 20 20 20 20 20 20 | 20 20 11 32 32 0d 0b 00 |        |  .22...|
|00002b00| 11 31 57 72 6f 6e 67 2e | 20 20 54 68 65 20 65 71 |.1Wrong.|  The eq|
|00002b10| 75 61 74 69 6f 6e 20 69 | 6e 20 73 74 61 6e 64 61 |uation i|n standa|
|00002b20| 72 64 20 66 6f 72 6d 20 | 69 73 20 31 38 11 33 78 |rd form |is 18.3x|
|00002b30| 20 20 11 31 2d 20 32 34 | 11 33 78 20 11 31 2b 20 |  .1- 24|.3x .1+ |
|00002b40| 38 20 3d 20 30 2e 20 20 | 57 69 74 68 20 11 33 61 |8 = 0.  |With .3a|
|00002b50| 20 11 31 3d 20 31 38 2c | 20 0d 0a 00 11 33 62 20 | .1= 18,| ....3b |
|00002b60| 11 31 3d 20 2d 32 34 2c | 20 61 6e 64 20 11 33 63 |.1= -24,| and .3c|
|00002b70| 20 11 31 3d 20 38 2c 20 | 77 65 20 73 6f 6c 76 65 | .1= 8, |we solve|
|00002b80| 20 61 73 20 66 6f 6c 6c | 6f 77 73 2e 12 30 0d 0a | as foll|ows..0..|
|00002b90| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00002ba0| 20 20 20 20 20 11 34 67 | 32 32 32 32 32 32 32 32 |     .4g|22222222|
|00002bb0| 32 32 32 32 32 32 32 32 | 0d 0b 00 20 20 20 20 20 |22222222|...     |
|00002bc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 66 20 |        |      f |
|00002bd0| 20 20 20 20 11 32 32 0d | 0b 00 20 20 20 20 20 20 |    .22.|..      |
|00002be0| 20 20 20 11 31 2d 28 2d | 32 34 29 20 11 34 2b 20 |   .1-(-|24) .4+ |
|00002bf0| 76 20 11 31 28 2d 32 34 | 29 20 20 2d 20 34 28 31 |v .1(-24|)  - 4(1|
|00002c00| 38 29 28 38 29 20 20 20 | 32 34 20 11 34 2b 20 11 |8)(8)   |24 .4+ .|
|00002c10| 31 30 20 20 20 32 34 20 | 20 20 32 0d 0b 00 20 20 |10   24 |  2...  |
|00002c20| 20 20 20 11 33 78 20 11 | 31 3d 20 11 34 32 32 32 |   .3x .|1= .4222|
|00002c30| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|00002c40| 32 32 32 32 32 32 32 32 | 32 20 11 31 3d 20 11 34 |22222222|2 .1= .4|
|00002c50| 32 32 32 32 32 32 20 11 | 31 3d 20 11 34 32 32 20 |222222 .|1= .422 |
|00002c60| 11 31 3d 20 11 34 32 0d | 0b 00 20 20 20 20 20 20 |.1= .42.|..      |
|00002c70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 31 |        |      .1|
|00002c80| 32 28 31 38 29 20 20 20 | 20 20 20 20 20 20 20 20 |2(18)   |        |
|00002c90| 20 20 20 20 20 20 33 36 | 20 20 20 20 20 33 36 20 |      36|     36 |
|00002ca0| 20 20 33 0d 0a 00 00 17 | 4f 03 0d 0b 00 52 69 67 |  3.....|O....Rig|
|00002cb0| 68 74 2e 20 20 54 68 65 | 20 65 71 75 61 74 69 6f |ht.  The| equatio|
|00002cc0| 6e 20 68 61 73 20 6f 6e | 65 20 73 6f 6c 75 74 69 |n has on|e soluti|
|00002cd0| 6f 6e 3a 20 32 2f 33 2e | 0d 0a 00 00 17 4f 03 0d |on: 2/3.|.....O..|
|00002ce0| 0b 00 57 72 6f 6e 67 2e | 20 20 54 68 65 20 67 69 |..Wrong.|  The gi|
|00002cf0| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 20 68 61 73 |ven equa|tion has|
|00002d00| 20 61 20 73 6f 6c 75 74 | 69 6f 6e 2e 20 20 54 72 | a solut|ion.  Tr|
|00002d10| 79 20 61 67 61 69 6e 2e | 0d 0a 00 00 17 4f 04 0d |y again.|.....O..|
|00002d20| 0b 00 57 72 6f 6e 67 2e | 20 20 43 68 65 63 6b 20 |..Wrong.|  Check |
|00002d30| 79 6f 75 72 20 63 61 6c | 63 75 6c 61 74 69 6f 6e |your cal|culation|
|00002d40| 73 20 6f 72 20 72 65 76 | 69 65 77 20 47 75 69 64 |s or rev|iew Guid|
|00002d50| 65 64 20 45 78 61 6d 70 | 6c 65 20 31 32 20 69 66 |ed Examp|le 12 if|
|00002d60| 20 79 6f 75 20 6e 65 65 | 64 20 0d 0a 00 6d 6f 72 | you nee|d ...mor|
|00002d70| 65 20 68 65 6c 70 2e 0d | 0a 00 00 17 4f 04 0d 0b |e help..|....O...|
|00002d80| 00 57 72 6f 6e 67 2e 20 | 20 43 68 65 63 6b 20 79 |.Wrong. | Check y|
|00002d90| 6f 75 72 20 63 61 6c 63 | 75 6c 61 74 69 6f 6e 73 |our calc|ulations|
|00002da0| 20 6f 72 20 72 65 76 69 | 65 77 20 47 75 69 64 65 | or revi|ew Guide|
|00002db0| 64 20 45 78 61 6d 70 6c | 65 20 31 32 20 69 66 20 |d Exampl|e 12 if |
|00002dc0| 79 6f 75 20 6e 65 65 64 | 20 0d 0a 00 6d 6f 72 65 |you need| ...more|
|00002dd0| 20 68 65 6c 70 2e 0d 0a | 00 00 17 4f 03 0d 0b 00 | help...|...O....|
|00002de0| 52 69 67 68 74 2e 20 20 | 59 6f 75 20 73 6f 6c 76 |Right.  |You solv|
|00002df0| 65 64 20 74 68 65 20 65 | 71 75 61 74 69 6f 6e 20 |ed the e|quation |
|00002e00| 75 73 69 6e 67 20 79 6f | 75 72 20 63 61 6c 63 75 |using yo|ur calcu|
|00002e10| 6c 61 74 6f 72 20 63 6f | 72 72 65 63 74 6c 79 2e |lator co|rrectly.|
|00002e20| 0d 0a 00 00 17 4f 04 0d | 0b 00 57 72 6f 6e 67 2e |.....O..|..Wrong.|
|00002e30| 20 20 43 68 65 63 6b 20 | 79 6f 75 72 20 63 61 6c |  Check |your cal|
|00002e40| 63 75 6c 61 74 69 6f 6e | 73 20 6f 72 20 72 65 76 |culation|s or rev|
|00002e50| 69 65 77 20 47 75 69 64 | 65 64 20 45 78 61 6d 70 |iew Guid|ed Examp|
|00002e60| 6c 65 20 31 32 20 69 66 | 20 79 6f 75 20 6e 65 65 |le 12 if| you nee|
|00002e70| 64 20 0d 0a 00 6d 6f 72 | 65 20 68 65 6c 70 2e 0d |d ...mor|e help..|
|00002e80| 0a 00 00 17 4f 03 0d 0b | 00 4e 6f 2e 20 20 11 33 |....O...|.No.  .3|
|00002e90| 78 20 11 31 3d 20 34 20 | 69 73 20 61 20 73 6f 6c |x .1= 4 |is a sol|
|00002ea0| 75 74 69 6f 6e 20 74 6f | 20 74 68 69 73 20 65 71 |ution to| this eq|
|00002eb0| 75 61 74 69 6f 6e 2c 20 | 62 75 74 20 11 33 78 20 |uation, |but .3x |
|00002ec0| 11 31 3d 20 2d 32 2f 33 | 20 69 73 20 61 6c 73 6f |.1= -2/3| is also|
|00002ed0| 20 61 20 73 6f 6c 75 74 | 69 6f 6e 2e 0d 0a 00 00 | a solut|ion.....|
|00002ee0| 17 4f 03 0d 0b 00 47 6f | 6f 64 20 6a 6f 62 2e 20 |.O....Go|od job. |
|00002ef0| 20 54 68 65 20 67 69 76 | 65 6e 20 65 71 75 61 74 | The giv|en equat|
|00002f00| 69 6f 6e 20 68 61 73 20 | 74 77 6f 20 73 6f 6c 75 |ion has |two solu|
|00002f10| 74 69 6f 6e 73 3a 20 34 | 20 61 6e 64 20 2d 32 2f |tions: 4| and -2/|
|00002f20| 33 2e 0d 0a 00 00 17 4f | 07 0d 0b 00 57 72 6f 6e |3......O|....Wron|
|00002f30| 67 2e 20 20 57 65 20 73 | 6f 6c 76 65 20 74 68 65 |g.  We s|olve the|
|00002f40| 20 65 71 75 61 74 69 6f | 6e 20 61 73 20 66 6f 6c | equatio|n as fol|
|00002f50| 6c 6f 77 73 2e 20 20 12 | 30 0d 0a 00 20 20 20 20 |lows.  .|0...    |
|00002f60| 20 20 20 20 20 20 20 20 | 20 20 20 20 11 32 32 0d |        |    .22.|
|00002f70| 0b 00 20 20 20 20 11 31 | 38 20 2b 20 31 30 11 33 |..    .1|8 + 10.3|
|00002f80| 78 20 11 31 2d 20 33 11 | 33 78 20 20 11 31 3d 20 |x .1- 3.|3x  .1= |
|00002f90| 30 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |0       |        |
|00002fa0| 20 20 20 20 20 20 20 11 | 32 12 31 47 69 76 65 6e |       .|2.1Given|
|00002fb0| 20 65 71 75 61 74 69 6f | 6e 12 30 0d 0a 00 20 20 | equatio|n.0...  |
|00002fc0| 11 31 28 34 20 2d 20 11 | 33 78 11 31 29 28 32 20 |.1(4 - .|3x.1)(2 |
|00002fd0| 2b 20 33 11 33 78 11 31 | 29 20 3d 20 30 20 20 20 |+ 3.3x.1|) = 0   |
|00002fe0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00002ff0| 20 20 20 11 32 12 31 46 | 61 63 74 6f 72 65 64 20 |   .2.1F|actored |
|00003000| 66 6f 72 6d 12 30 0d 0a | 00 20 20 20 20 20 20 20 |form.0..|.       |
|00003010| 20 20 20 20 20 11 31 34 | 20 2d 20 11 33 78 20 11 |     .14| - .3x .|
|00003020| 31 3d 20 30 20 20 20 11 | 34 35 35 36 20 20 20 11 |1= 0   .|4556   .|
|00003030| 33 78 20 11 31 3d 20 34 | 20 20 20 20 20 20 20 20 |3x .1= 4|        |
|00003040| 11 32 12 31 53 65 74 20 | 31 73 74 20 66 61 63 74 |.2.1Set |1st fact|
|00003050| 6f 72 20 65 71 75 61 6c | 20 74 6f 20 30 12 30 0d |or equal| to 0.0.|
|00003060| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 11 31 32 |..      |     .12|
|00003070| 20 2b 20 33 11 33 78 20 | 11 31 3d 20 30 20 20 20 | + 3.3x |.1= 0   |
|00003080| 11 34 35 35 36 20 20 20 | 11 33 78 20 11 31 3d 20 |.4556   |.3x .1= |
|00003090| 2d 32 2f 33 20 20 20 20 | 20 11 32 12 31 53 65 74 |-2/3    | .2.1Set|
|000030a0| 20 32 6e 64 20 66 61 63 | 74 6f 72 20 65 71 75 61 | 2nd fac|tor equa|
|000030b0| 6c 20 74 6f 20 30 12 33 | 0d 0a 00 00 17 4f 03 0d |l to 0.3|.....O..|
|000030c0| 0b 00 4e 6f 2e 20 20 11 | 33 78 20 11 31 3d 20 2d |..No.  .|3x .1= -|
|000030d0| 32 2f 33 20 69 73 20 61 | 20 73 6f 6c 75 74 69 6f |2/3 is a| solutio|
|000030e0| 6e 20 74 6f 20 74 68 69 | 73 20 65 71 75 61 74 69 |n to thi|s equati|
|000030f0| 6f 6e 2c 20 62 75 74 20 | 11 33 78 20 11 31 3d 20 |on, but |.3x .1= |
|00003100| 34 20 69 73 20 61 6c 73 | 6f 20 61 20 73 6f 6c 75 |4 is als|o a solu|
|00003110| 74 69 6f 6e 2e 0d 0a 00 | 00 17 4f 03 0d 0b 00 52 |tion....|..O....R|
|00003120| 69 67 68 74 2e 20 20 54 | 68 65 20 67 69 76 65 6e |ight.  T|he given|
|00003130| 20 65 71 75 61 74 69 6f | 6e 20 68 61 73 20 74 77 | equatio|n has tw|
|00003140| 6f 20 73 6f 6c 75 74 69 | 6f 6e 73 3a 20 35 2f 33 |o soluti|ons: 5/3|
|00003150| 20 61 6e 64 20 2d 35 2f | 33 2e 0d 0a 00 00 17 4f | and -5/|3......O|
|00003160| 0b 0d 0b 00 57 72 6f 6e | 67 2e 20 20 57 65 20 73 |....Wron|g.  We s|
|00003170| 6f 6c 76 65 20 74 68 65 | 20 67 69 76 65 6e 20 65 |olve the| given e|
|00003180| 71 75 61 74 69 6f 6e 20 | 61 73 20 66 6f 6c 6c 6f |quation |as follo|
|00003190| 77 73 2e 20 20 28 4e 6f | 74 65 20 74 68 61 74 20 |ws.  (No|te that |
|000031a0| 77 65 20 63 6f 75 6c 64 | 20 68 61 76 65 20 0d 0a |we could| have ..|
|000031b0| 00 63 68 6f 73 65 6e 20 | 61 6e 6f 74 68 65 72 20 |.chosen |another |
|000031c0| 6d 65 74 68 6f 64 20 74 | 6f 20 73 6f 6c 76 65 20 |method t|o solve |
|000031d0| 74 68 69 73 20 65 71 75 | 61 74 69 6f 6e 2c 20 73 |this equ|ation, s|
|000031e0| 75 63 68 20 61 73 20 62 | 79 20 66 61 63 74 6f 72 |uch as b|y factor|
|000031f0| 69 6e 67 2e 29 12 30 0d | 0a 00 20 20 20 20 20 20 |ing.).0.|..      |
|00003200| 20 11 32 32 0d 0b 00 20 | 20 20 20 20 11 31 39 11 | .22... |    .19.|
|00003210| 33 78 20 20 11 31 2d 20 | 32 35 20 3d 20 30 20 20 |3x  .1- |25 = 0  |
|00003220| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 47 69 76 |        | .2.1Giv|
|00003230| 65 6e 20 65 71 75 61 74 | 69 6f 6e 12 30 0d 0a 00 |en equat|ion.0...|
|00003240| 20 20 20 20 20 20 20 20 | 20 20 20 20 32 0d 0b 00 |        |    2...|
|00003250| 20 20 20 20 20 20 20 20 | 20 20 11 31 39 11 33 78 |        |  .19.3x|
|00003260| 20 20 11 31 3d 20 32 35 | 20 20 20 20 20 20 20 20 |  .1= 25|        |
|00003270| 20 20 11 32 12 31 41 64 | 64 20 32 35 20 74 6f 20 |  .2.1Ad|d 25 to |
|00003280| 62 6f 74 68 20 73 69 64 | 65 73 12 30 0d 0a 00 20 |both sid|es.0... |
|00003290| 20 20 20 20 20 20 20 20 | 20 20 20 32 20 20 20 11 |        |   2   .|
|000032a0| 31 32 35 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |125...  |        |
|000032b0| 20 11 33 78 20 20 11 31 | 3d 20 11 34 32 32 20 20 | .3x  .1|= .422  |
|000032c0| 20 20 20 20 20 20 20 20 | 11 32 12 31 44 69 76 69 |        |.2.1Divi|
|000032d0| 64 65 20 62 79 20 39 12 | 30 0d 0b 00 20 20 20 20 |de by 9.|0...    |
|000032e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 31 39 |        |     .19|
|000032f0| 0d 0a 00 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |...     |        |
|00003300| 20 20 20 20 35 0d 0b 00 | 20 20 20 20 20 20 20 20 |    5...|        |
|00003310| 20 20 20 20 11 33 78 20 | 11 31 3d 20 11 34 2b 32 |    .3x |.1= .4+2|
|00003320| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 45 78 |        |  .2.1Ex|
|00003330| 74 72 61 63 74 20 73 71 | 75 61 72 65 20 72 6f 6f |tract sq|uare roo|
|00003340| 74 73 12 30 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |ts.0... |        |
|00003350| 20 20 20 20 20 20 20 20 | 11 31 33 0d 0a 00 00 17 |        |.13.....|
|00003360| 4f 0b 0d 0b 00 57 72 6f | 6e 67 2e 20 20 57 65 20 |O....Wro|ng.  We |
|00003370| 73 6f 6c 76 65 20 74 68 | 65 20 67 69 76 65 6e 20 |solve th|e given |
|00003380| 65 71 75 61 74 69 6f 6e | 20 61 73 20 66 6f 6c 6c |equation| as foll|
|00003390| 6f 77 73 2e 20 20 28 4e | 6f 74 65 20 74 68 61 74 |ows.  (N|ote that|
|000033a0| 20 77 65 20 63 6f 75 6c | 64 20 68 61 76 65 20 0d | we coul|d have .|
|000033b0| 0a 00 63 68 6f 73 65 6e | 20 61 6e 6f 74 68 65 72 |..chosen| another|
|000033c0| 20 6d 65 74 68 6f 64 20 | 74 6f 20 73 6f 6c 76 65 | method |to solve|
|000033d0| 20 74 68 69 73 20 65 71 | 75 61 74 69 6f 6e 2c 20 | this eq|uation, |
|000033e0| 73 75 63 68 20 61 73 20 | 62 79 20 66 61 63 74 6f |such as |by facto|
|000033f0| 72 69 6e 67 2e 29 12 30 | 0d 0a 00 20 20 20 20 20 |ring.).0|...     |
|00003400| 20 20 11 32 32 0d 0b 00 | 20 20 20 20 20 11 31 39 |  .22...|     .19|
|00003410| 11 33 78 20 20 11 31 2d | 20 32 35 20 3d 20 30 20 |.3x  .1-| 25 = 0 |
|00003420| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 47 69 |        |  .2.1Gi|
|00003430| 76 65 6e 20 65 71 75 61 | 74 69 6f 6e 12 30 0d 0a |ven equa|tion.0..|
|00003440| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 32 0d 0b |.       |     2..|
|00003450| 00 20 20 20 20 20 20 20 | 20 20 20 11 31 39 11 33 |.       |   .19.3|
|00003460| 78 20 20 11 31 3d 20 32 | 35 20 20 20 20 20 20 20 |x  .1= 2|5       |
|00003470| 20 20 20 11 32 12 31 41 | 64 64 20 32 35 20 74 6f |   .2.1A|dd 25 to|
|00003480| 20 62 6f 74 68 20 73 69 | 64 65 73 12 30 0d 0a 00 | both si|des.0...|
|00003490| 20 20 20 20 20 20 20 20 | 20 20 20 20 32 20 20 20 |        |    2   |
|000034a0| 11 31 32 35 0d 0b 00 20 | 20 20 20 20 20 20 20 20 |.125... |        |
|000034b0| 20 20 11 33 78 20 20 11 | 31 3d 20 11 34 32 32 20 |  .3x  .|1= .422 |
|000034c0| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 44 69 76 |        | .2.1Div|
|000034d0| 69 64 65 20 62 79 20 39 | 12 30 0d 0b 00 20 20 20 |ide by 9|.0...   |
|000034e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 31 |        |      .1|
|000034f0| 39 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |9...    |        |
|00003500| 20 20 20 20 20 35 0d 0b | 00 20 20 20 20 20 20 20 |     5..|.       |
|00003510| 20 20 20 20 20 11 33 78 | 20 11 31 3d 20 11 34 2b |     .3x| .1= .4+|
|00003520| 32 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 45 |2       |   .2.1E|
|00003530| 78 74 72 61 63 74 20 73 | 71 75 61 72 65 20 72 6f |xtract s|quare ro|
|00003540| 6f 74 73 12 30 0d 0b 00 | 20 20 20 20 20 20 20 20 |ots.0...|        |
|00003550| 20 20 20 20 20 20 20 20 | 20 11 31 33 0d 0a 00 00 |        | .13....|
|00003560| 17 4f 0b 0d 0b 00 57 72 | 6f 6e 67 2e 20 20 57 65 |.O....Wr|ong.  We|
|00003570| 20 73 6f 6c 76 65 20 74 | 68 65 20 67 69 76 65 6e | solve t|he given|
|00003580| 20 65 71 75 61 74 69 6f | 6e 20 61 73 20 66 6f 6c | equatio|n as fol|
|00003590| 6c 6f 77 73 2e 20 20 28 | 4e 6f 74 65 20 74 68 61 |lows.  (|Note tha|
|000035a0| 74 20 77 65 20 63 6f 75 | 6c 64 20 68 61 76 65 20 |t we cou|ld have |
|000035b0| 0d 0a 00 63 68 6f 73 65 | 6e 20 61 6e 6f 74 68 65 |...chose|n anothe|
|000035c0| 72 20 6d 65 74 68 6f 64 | 20 74 6f 20 73 6f 6c 76 |r method| to solv|
|000035d0| 65 20 74 68 69 73 20 65 | 71 75 61 74 69 6f 6e 2c |e this e|quation,|
|000035e0| 20 73 75 63 68 20 61 73 | 20 62 79 20 66 61 63 74 | such as| by fact|
|000035f0| 6f 72 69 6e 67 2e 29 12 | 30 0d 0a 00 20 20 20 20 |oring.).|0...    |
|00003600| 20 20 20 11 32 32 0d 0b | 00 20 20 20 20 20 11 31 |   .22..|.     .1|
|00003610| 39 11 33 78 20 20 11 31 | 2d 20 32 35 20 3d 20 30 |9.3x  .1|- 25 = 0|
|00003620| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 12 31 47 |        |   .2.1G|
|00003630| 69 76 65 6e 20 65 71 75 | 61 74 69 6f 6e 12 30 0d |iven equ|ation.0.|
|00003640| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 32 0d |..      |      2.|
|00003650| 0b 00 20 20 20 20 20 20 | 20 20 20 20 11 31 39 11 |..      |    .19.|
|00003660| 33 78 20 20 11 31 3d 20 | 32 35 20 20 20 20 20 20 |3x  .1= |25      |
|00003670| 20 20 20 20 11 32 12 31 | 41 64 64 20 32 35 20 74 |    .2.1|Add 25 t|
|00003680| 6f 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0a |o both s|ides.0..|
|00003690| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 32 20 20 |.       |     2  |
|000036a0| 20 11 31 32 35 0d 0b 00 | 20 20 20 20 20 20 20 20 | .125...|        |
|000036b0| 20 20 20 11 33 78 20 20 | 11 31 3d 20 11 34 32 32 |   .3x  |.1= .422|
|000036c0| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 44 69 |        |  .2.1Di|
|000036d0| 76 69 64 65 20 62 79 20 | 39 12 30 0d 0b 00 20 20 |vide by |9.0...  |
|000036e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|000036f0| 31 39 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |19...   |        |
|00003700| 20 20 20 20 20 20 35 0d | 0b 00 20 20 20 20 20 20 |      5.|..      |
|00003710| 20 20 20 20 20 20 11 33 | 78 20 11 31 3d 20 11 34 |      .3|x .1= .4|
|00003720| 2b 32 20 20 20 20 20 20 | 20 20 20 20 11 32 12 31 |+2      |    .2.1|
|00003730| 45 78 74 72 61 63 74 20 | 73 71 75 61 72 65 20 72 |Extract |square r|
|00003740| 6f 6f 74 73 12 30 0d 0b | 00 20 20 20 20 20 20 20 |oots.0..|.       |
|00003750| 20 20 20 20 20 20 20 20 | 20 20 11 31 33 0d 0a 00 |        |  .13...|
|00003760| 00 17 4f 07 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 57 |..O....W|rong.  W|
|00003770| 65 20 63 61 6e 20 73 6f | 6c 76 65 20 74 68 65 20 |e can so|lve the |
|00003780| 67 69 76 65 6e 20 65 71 | 75 61 74 69 6f 6e 20 61 |given eq|uation a|
|00003790| 73 20 66 6f 6c 6c 6f 77 | 73 2e 12 30 0d 0a 00 20 |s follow|s..0... |
|000037a0| 20 20 20 20 20 20 20 20 | 20 20 20 11 32 32 0d 0b |        |   .22..|
|000037b0| 00 20 20 20 20 20 11 31 | 28 11 33 78 20 11 31 2b |.     .1|(.3x .1+|
|000037c0| 20 34 29 20 20 3d 20 36 | 34 20 20 20 20 20 20 20 | 4)  = 6|4       |
|000037d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000037e0| 20 20 20 20 20 20 20 20 | 11 32 12 31 47 69 76 65 |        |.2.1Give|
|000037f0| 6e 20 65 71 75 61 74 69 | 6f 6e 12 30 0d 0a 00 20 |n equati|on.0... |
|00003800| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 2b 20 34 |       .|3x .1+ 4|
|00003810| 20 3d 20 11 34 2b 11 31 | 38 20 20 20 20 20 20 20 | = .4+.1|8       |
|00003820| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003830| 20 20 20 20 20 20 20 20 | 11 32 12 31 45 78 74 72 |        |.2.1Extr|
|00003840| 61 63 74 20 73 71 75 61 | 72 65 20 72 6f 6f 74 73 |act squa|re roots|
|00003850| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|00003860| 20 11 33 78 20 11 31 3d | 20 2d 34 20 2b 20 38 20 | .3x .1=| -4 + 8 |
|00003870| 20 20 6f 72 20 20 20 11 | 33 78 20 11 31 3d 20 2d |  or   .|3x .1= -|
|00003880| 34 20 2d 20 38 20 20 20 | 20 20 20 20 20 20 11 32 |4 - 8   |      .2|
|00003890| 12 31 53 75 62 74 72 61 | 63 74 20 34 20 66 72 6f |.1Subtra|ct 4 fro|
|000038a0| 6d 20 62 6f 74 68 20 73 | 69 64 65 73 12 30 0d 0a |m both s|ides.0..|
|000038b0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 11 33 78 |.       |     .3x|
|000038c0| 20 11 31 3d 20 34 20 20 | 20 20 20 20 20 20 6f 72 | .1= 4  |      or|
|000038d0| 20 20 20 11 33 78 20 11 | 31 3d 20 2d 31 32 20 20 |   .3x .|1= -12  |
|000038e0| 20 20 20 20 20 20 20 20 | 20 20 11 32 12 31 53 6f |        |  .2.1So|
|000038f0| 6c 76 65 20 66 6f 72 20 | 78 12 30 0d 0a 00 00 17 |lve for |x.0.....|
|00003900| 4f 07 0d 0b 00 4e 6f 2e | 20 20 54 68 69 73 20 69 |O....No.|  This i|
|00003910| 73 20 6f 6e 6c 79 20 6f | 6e 65 20 6f 66 20 74 68 |s only o|ne of th|
|00003920| 65 20 73 6f 6c 75 74 69 | 6f 6e 73 2e 20 20 57 65 |e soluti|ons.  We|
|00003930| 20 73 6f 6c 76 65 20 61 | 73 20 66 6f 6c 6c 6f 77 | solve a|s follow|
|00003940| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |s..0... |        |
|00003950| 20 20 20 11 32 32 0d 0b | 00 20 20 20 20 20 11 31 |   .22..|.     .1|
|00003960| 28 11 33 78 20 11 31 2b | 20 34 29 20 20 3d 20 36 |(.3x .1+| 4)  = 6|
|00003970| 34 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |4       |        |
|00003980| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003990| 11 32 12 31 47 69 76 65 | 6e 20 65 71 75 61 74 69 |.2.1Give|n equati|
|000039a0| 6f 6e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 11 |on.0... |       .|
|000039b0| 33 78 20 11 31 2b 20 34 | 20 3d 20 11 34 2b 11 31 |3x .1+ 4| = .4+.1|
|000039c0| 38 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |8       |        |
|000039d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000039e0| 11 32 12 31 45 78 74 72 | 61 63 74 20 73 71 75 61 |.2.1Extr|act squa|
|000039f0| 72 65 20 72 6f 6f 74 73 | 12 30 0d 0a 00 20 20 20 |re roots|.0...   |
|00003a00| 20 20 20 20 20 20 20 20 | 20 11 33 78 20 11 31 3d |        | .3x .1=|
|00003a10| 20 2d 34 20 2b 20 38 20 | 20 20 6f 72 20 20 20 11 | -4 + 8 |  or   .|
|00003a20| 33 78 20 11 31 3d 20 2d | 34 20 2d 20 38 20 20 20 |3x .1= -|4 - 8   |
|00003a30| 20 20 20 20 20 20 11 32 | 12 31 53 75 62 74 72 61 |      .2|.1Subtra|
|00003a40| 63 74 20 34 20 66 72 6f | 6d 20 62 6f 74 68 20 73 |ct 4 fro|m both s|
|00003a50| 69 64 65 73 12 30 0d 0a | 00 20 20 20 20 20 20 20 |ides.0..|.       |
|00003a60| 20 20 20 20 20 11 33 78 | 20 11 31 3d 20 34 20 20 |     .3x| .1= 4  |
|00003a70| 20 20 20 20 20 20 6f 72 | 20 20 20 11 33 78 20 11 |      or|   .3x .|
|00003a80| 31 3d 20 2d 31 32 20 20 | 20 20 20 20 20 20 20 20 |1= -12  |        |
|00003a90| 20 20 11 32 12 31 53 6f | 6c 76 65 20 66 6f 72 20 |  .2.1So|lve for |
|00003aa0| 78 12 30 0d 0a 00 00 17 | 4f 07 0d 0b 00 57 72 6f |x.0.....|O....Wro|
|00003ab0| 6e 67 2e 20 20 57 65 20 | 63 61 6e 20 73 6f 6c 76 |ng.  We |can solv|
|00003ac0| 65 20 74 68 65 20 67 69 | 76 65 6e 20 65 71 75 61 |e the gi|ven equa|
|00003ad0| 74 69 6f 6e 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |tion as |follows.|
|00003ae0| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|00003af0| 20 11 32 32 0d 0b 00 20 | 20 20 20 20 11 31 28 11 | .22... |    .1(.|
|00003b00| 33 78 20 11 31 2b 20 34 | 29 20 20 3d 20 36 34 20 |3x .1+ 4|)  = 64 |
|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 20 11 32 |        |      .2|
|00003b30| 12 31 47 69 76 65 6e 20 | 65 71 75 61 74 69 6f 6e |.1Given |equation|
|00003b40| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 11 33 78 |.0...   |     .3x|
|00003b50| 20 11 31 2b 20 34 20 3d | 20 11 34 2b 11 31 38 20 | .1+ 4 =| .4+.18 |
|00003b60| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003b70| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 32 |        |      .2|
|00003b80| 12 31 45 78 74 72 61 63 | 74 20 73 71 75 61 72 65 |.1Extrac|t square|
|00003b90| 20 72 6f 6f 74 73 12 30 | 0d 0a 00 20 20 20 20 20 | roots.0|...     |
|00003ba0| 20 20 20 20 20 20 20 11 | 33 78 20 11 31 3d 20 2d |       .|3x .1= -|
|00003bb0| 34 20 2b 20 38 20 20 20 | 6f 72 20 20 20 11 33 78 |4 + 8   |or   .3x|
|00003bc0| 20 11 31 3d 20 2d 34 20 | 2d 20 38 20 20 20 20 20 | .1= -4 |- 8     |
|00003bd0| 20 20 20 20 11 32 12 31 | 53 75 62 74 72 61 63 74 |    .2.1|Subtract|
|00003be0| 20 34 20 66 72 6f 6d 20 | 62 6f 74 68 20 73 69 64 | 4 from |both sid|
|00003bf0| 65 73 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |es.0... |        |
|00003c00| 20 20 20 11 33 78 20 11 | 31 3d 20 34 20 20 20 20 |   .3x .|1= 4    |
|00003c10| 20 20 20 20 6f 72 20 20 | 20 11 33 78 20 11 31 3d |    or  | .3x .1=|
|00003c20| 20 2d 31 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | -12    |        |
|00003c30| 11 32 12 31 53 6f 6c 76 | 65 20 66 6f 72 20 78 12 |.2.1Solv|e for x.|
|00003c40| 30 0d 0a 00 00 17 4f 03 | 0d 0b 00 52 69 67 68 74 |0.....O.|...Right|
|00003c50| 2e 20 20 54 68 65 20 67 | 69 76 65 6e 20 65 71 75 |.  The g|iven equ|
|00003c60| 61 74 69 6f 6e 20 68 61 | 73 20 74 77 6f 20 73 6f |ation ha|s two so|
|00003c70| 6c 75 74 69 6f 6e 73 3a | 20 34 20 61 6e 64 20 2d |lutions:| 4 and -|
|00003c80| 31 32 2e 0d 0a 00 00 17 | 4f 06 20 20 20 20 20 20 |12......|O.      |
|00003c90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003ca0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003cb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003cc0| 20 20 20 20 20 20 20 20 | 11 32 32 0d 0b 00 11 31 |        |.22....1|
|00003cd0| 4e 6f 74 20 71 75 69 74 | 65 2e 20 20 59 6f 75 20 |Not quit|e.  You |
|00003ce0| 68 61 76 65 20 79 6f 75 | 72 20 73 69 67 6e 73 20 |have you|r signs |
|00003cf0| 6d 65 73 73 65 64 20 75 | 70 2e 20 20 53 69 6e 63 |messed u|p.  Sinc|
|00003d00| 65 20 32 37 20 2d 20 33 | 11 33 78 20 11 31 3d 20 |e 27 - 3|.3x .1= |
|00003d10| 32 11 33 78 20 20 11 31 | 63 61 6e 20 62 65 20 0d |2.3x  .1|can be .|
|00003d20| 0a 00 0d 0b 00 66 61 63 | 74 6f 72 65 64 20 61 73 |.....fac|tored as|
|00003d30| 20 28 32 11 33 78 20 11 | 31 2b 20 39 29 28 11 33 | (2.3x .|1+ 9)(.3|
|00003d40| 78 20 11 31 2d 20 33 29 | 20 3d 20 30 20 61 66 74 |x .1- 3)| = 0 aft|
|00003d50| 65 72 20 62 65 69 6e 67 | 20 77 72 69 74 74 65 6e |er being| written|
|00003d60| 20 69 6e 20 73 74 61 6e | 64 61 72 64 20 66 6f 72 | in stan|dard for|
|00003d70| 6d 2c 20 77 65 20 0d 0a | 00 20 20 20 20 20 20 20 |m, we ..|.       |
|00003d80| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003d90| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 39 0d 0b |        |     9..|
|00003da0| 00 63 61 6e 20 73 65 65 | 20 74 68 61 74 20 74 68 |.can see| that th|
|00003db0| 65 20 73 6f 6c 75 74 69 | 6f 6e 73 20 61 72 65 20 |e soluti|ons are |
|00003dc0| 11 33 78 20 11 31 3d 20 | 2d 11 34 32 20 11 31 61 |.3x .1= |-.42 .1a|
|00003dd0| 6e 64 20 11 33 78 20 11 | 31 3d 20 33 2e 0d 0b 00 |nd .3x .|1= 3....|
|00003de0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003df0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003e00| 20 20 20 20 32 0d 0a 00 | 00 17 4f 03 0d 0b 00 56 |    2...|..O....V|
|00003e10| 65 72 79 20 67 6f 6f 64 | 2e 20 20 54 68 65 20 73 |ery good|.  The s|
|00003e20| 6f 6c 75 74 69 6f 6e 73 | 20 6f 66 20 74 68 65 20 |olutions| of the |
|00003e30| 67 69 76 65 6e 20 65 71 | 75 61 74 69 6f 6e 20 61 |given eq|uation a|
|00003e40| 72 65 20 2d 39 2f 32 20 | 61 6e 64 20 33 2e 0d 0a |re -9/2 |and 3...|
|00003e50| 00 00 17 4f 06 0d 0b 00 | 4e 6f 74 20 71 75 69 74 |...O....|Not quit|
|00003e60| 65 2e 20 20 4f 6e 65 20 | 6f 66 20 74 68 65 20 73 |e.  One |of the s|
|00003e70| 6f 6c 75 74 69 6f 6e 73 | 20 69 73 20 33 2c 20 62 |olutions| is 3, b|
|00003e80| 75 74 20 74 68 65 20 6f | 74 68 65 72 20 73 6f 6c |ut the o|ther sol|
|00003e90| 75 74 69 6f 6e 20 69 73 | 20 2d 39 2f 32 2e 20 0d |ution is| -9/2. .|
|00003ea0| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|00003eb0| 20 20 20 20 11 32 32 0d | 0b 00 11 31 53 69 6e 63 |    .22.|...1Sinc|
|00003ec0| 65 20 32 37 20 2d 20 33 | 11 33 78 20 11 31 3d 20 |e 27 - 3|.3x .1= |
|00003ed0| 32 11 33 78 20 20 11 31 | 63 61 6e 20 62 65 20 66 |2.3x  .1|can be f|
|00003ee0| 61 63 74 6f 72 65 64 20 | 61 73 20 28 32 11 33 78 |actored |as (2.3x|
|00003ef0| 20 11 31 2b 20 39 29 28 | 11 33 78 20 11 31 2d 20 | .1+ 9)(|.3x .1- |
|00003f00| 33 29 20 3d 20 30 20 61 | 66 74 65 72 20 62 65 69 |3) = 0 a|fter bei|
|00003f10| 6e 67 20 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |ng ...  |        |
|00003f20| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003f30| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003f40| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003f50| 20 20 20 39 0d 0b 00 77 | 72 69 74 74 65 6e 20 69 |   9...w|ritten i|
|00003f60| 6e 20 73 74 61 6e 64 61 | 72 64 20 66 6f 72 6d 2c |n standa|rd form,|
|00003f70| 20 77 65 20 73 65 65 20 | 74 68 61 74 20 74 68 65 | we see |that the|
|00003f80| 20 73 6f 6c 75 74 69 6f | 6e 73 20 61 72 65 20 11 | solutio|ns are .|
|00003f90| 33 78 20 11 31 3d 20 2d | 11 34 32 20 11 31 61 6e |3x .1= -|.42 .1an|
|00003fa0| 64 20 11 33 78 20 11 31 | 3d 20 33 2e 0d 0b 00 20 |d .3x .1|= 3.... |
|00003fb0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003fc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003fd0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00003fe0| 20 20 20 20 20 20 20 20 | 20 20 20 20 32 0d 0a 00 |        |    2...|
|00003ff0| 00 17 4f 06 0d 0b 00 4e | 6f 74 20 71 75 69 74 65 |..O....N|ot quite|
|00004000| 2e 20 20 4f 6e 65 20 6f | 66 20 74 68 65 20 73 6f |.  One o|f the so|
|00004010| 6c 75 74 69 6f 6e 73 20 | 69 73 20 2d 39 2f 32 2c |lutions |is -9/2,|
|00004020| 20 62 75 74 20 74 68 65 | 20 6f 74 68 65 72 20 73 | but the| other s|
|00004030| 6f 6c 75 74 69 6f 6e 20 | 69 73 20 33 2e 20 0d 0a |olution |is 3. ..|
|00004040| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00004050| 20 20 20 11 32 32 0d 0b | 00 11 31 53 69 6e 63 65 |   .22..|..1Since|
|00004060| 20 32 37 20 2d 20 33 11 | 33 78 20 11 31 3d 20 32 | 27 - 3.|3x .1= 2|
|00004070| 11 33 78 20 20 11 31 63 | 61 6e 20 62 65 20 66 61 |.3x  .1c|an be fa|
|00004080| 63 74 6f 72 65 64 20 61 | 73 20 28 32 11 33 78 20 |ctored a|s (2.3x |
|00004090| 11 31 2b 20 39 29 28 11 | 33 78 20 11 31 2d 20 33 |.1+ 9)(.|3x .1- 3|
|000040a0| 29 20 3d 20 30 20 61 66 | 74 65 72 20 62 65 69 6e |) = 0 af|ter bein|
|000040b0| 67 20 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |g ...   |        |
|000040c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000040d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000040e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000040f0| 20 20 39 0d 0b 00 77 72 | 69 74 74 65 6e 20 69 6e |  9...wr|itten in|
|00004100| 20 73 74 61 6e 64 61 72 | 64 20 66 6f 72 6d 2c 20 | standar|d form, |
|00004110| 77 65 20 73 65 65 20 74 | 68 61 74 20 74 68 65 20 |we see t|hat the |
|00004120| 73 6f 6c 75 74 69 6f 6e | 73 20 61 72 65 20 11 33 |solution|s are .3|
|00004130| 78 20 11 31 3d 20 2d 11 | 34 32 20 11 31 61 6e 64 |x .1= -.|42 .1and|
|00004140| 20 11 33 78 20 11 31 3d | 20 33 2e 0d 0b 00 20 20 | .3x .1=| 3....  |
|00004150| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004160| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004170| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004180| 20 20 20 20 20 20 20 20 | 20 20 20 32 0d 0a 00 00 |        |   2....|
|00004190| 17 4f 04 0d 0b 00 49 6e | 63 6f 72 72 65 63 74 2e |.O....In|correct.|
|000041a0| 20 20 59 6f 75 20 73 68 | 6f 75 6c 64 20 73 6f 6c |  You sh|ould sol|
|000041b0| 76 65 20 74 68 69 73 20 | 65 71 75 61 74 69 6f 6e |ve this |equation|
|000041c0| 20 62 79 20 63 6f 6d 70 | 6c 65 74 69 6e 67 20 74 | by comp|leting t|
|000041d0| 68 65 20 73 71 75 61 72 | 65 2e 20 20 54 72 79 20 |he squar|e.  Try |
|000041e0| 0d 0a 00 61 67 61 69 6e | 2e 0d 0a 00 00 17 4f 04 |...again|......O.|
|000041f0| 0d 0b 00 49 6e 63 6f 72 | 72 65 63 74 2e 20 20 59 |...Incor|rect.  Y|
|00004200| 6f 75 20 73 68 6f 75 6c | 64 20 73 6f 6c 76 65 20 |ou shoul|d solve |
|00004210| 74 68 69 73 20 65 71 75 | 61 74 69 6f 6e 20 62 79 |this equ|ation by|
|00004220| 20 63 6f 6d 70 6c 65 74 | 69 6e 67 20 74 68 65 20 | complet|ing the |
|00004230| 73 71 75 61 72 65 2e 20 | 20 54 72 79 0d 0a 00 61 |square. | Try...a|
|00004240| 67 61 69 6e 2e 0d 0a 00 | 00 17 4f 04 0d 0b 00 49 |gain....|..O....I|
|00004250| 6e 63 6f 72 72 65 63 74 | 2e 20 20 59 6f 75 20 73 |ncorrect|.  You s|
|00004260| 68 6f 75 6c 64 20 73 6f | 6c 76 65 20 74 68 69 73 |hould so|lve this|
|00004270| 20 65 71 75 61 74 69 6f | 6e 20 62 79 20 63 6f 6d | equatio|n by com|
|00004280| 70 6c 65 74 69 6e 67 20 | 74 68 65 20 73 71 75 61 |pleting |the squa|
|00004290| 72 65 2e 20 20 54 72 79 | 0d 0a 00 61 67 61 69 6e |re.  Try|...again|
|000042a0| 2e 0d 0a 00 00 17 4f 03 | 0d 0b 00 56 65 72 79 20 |......O.|...Very |
|000042b0| 67 6f 6f 64 2e 20 20 0d | 0a 00 00 17 4f 03 0d 0b |good.  .|....O...|
|000042c0| 00 56 65 72 79 20 67 6f | 6f 64 2e 20 20 54 68 65 |.Very go|od.  The|
|000042d0| 20 73 6f 6c 75 74 69 6f | 6e 20 6f 66 20 74 68 65 | solutio|n of the|
|000042e0| 20 67 69 76 65 6e 20 65 | 71 75 61 74 69 6f 6e 20 | given e|quation |
|000042f0| 69 73 20 33 2f 32 2e 0d | 0a 00 00 17 4f 0b 0d 0b |is 3/2..|....O...|
|00004300| 00 57 72 6f 6e 67 2e 20 | 20 57 65 20 63 61 6e 20 |.Wrong. | We can |
|00004310| 73 6f 6c 76 65 20 74 68 | 65 20 67 69 76 65 6e 20 |solve th|e given |
|00004320| 65 71 75 61 74 69 6f 6e | 20 61 73 20 66 6f 6c 6c |equation| as foll|
|00004330| 6f 77 73 2e 20 12 30 0d | 0a 00 20 20 20 20 20 20 |ows. .0.|..      |
|00004340| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004350| 20 20 20 11 32 32 20 20 | 20 20 32 0d 0b 00 20 20 |   .22  |  2...  |
|00004360| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004370| 11 31 28 11 33 78 20 11 | 31 2d 20 33 29 20 20 3d |.1(.3x .|1- 3)  =|
|00004380| 20 11 33 78 20 20 20 20 | 20 20 20 20 20 20 20 11 | .3x    |       .|
|00004390| 32 12 31 47 69 76 65 6e | 20 65 71 75 61 74 69 6f |2.1Given| equatio|
|000043a0| 6e 12 30 0d 0a 00 20 20 | 20 20 20 20 20 20 20 20 |n.0...  |        |
|000043b0| 20 20 20 20 20 20 20 20 | 20 20 32 20 20 20 20 32 |        |  2    2|
|000043c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 12 |        |       .|
|000043d0| 31 20 20 20 20 20 20 20 | 20 20 20 32 20 20 20 20 |1       |   2    |
|000043e0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 12 |        |       .|
|000043f0| 30 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |0...    |        |
|00004400| 20 11 31 28 11 33 78 20 | 11 31 2d 20 33 29 20 20 | .1(.3x |.1- 3)  |
|00004410| 2d 20 11 33 78 20 20 11 | 31 3d 20 30 20 20 20 20 |- .3x  .|1= 0    |
|00004420| 20 20 20 20 20 20 20 11 | 32 12 31 53 75 62 74 72 |       .|2.1Subtr|
|00004430| 61 63 74 20 78 20 20 66 | 72 6f 6d 20 62 6f 74 68 |act x  f|rom both|
|00004440| 20 73 69 64 65 73 12 30 | 0d 0a 00 0d 0b 00 11 31 | sides.0|.......1|
|00004450| 5b 28 11 33 78 20 11 31 | 2d 20 33 29 20 2b 20 11 |[(.3x .1|- 3) + .|
|00004460| 33 78 11 31 5d 5b 28 11 | 33 78 20 11 31 2d 20 33 |3x.1][(.|3x .1- 3|
|00004470| 29 20 2d 20 11 33 78 11 | 31 5d 20 3d 20 30 20 20 |) - .3x.|1] = 0  |
|00004480| 20 20 20 20 20 20 20 20 | 20 11 32 12 31 46 61 63 |        | .2.1Fac|
|00004490| 74 6f 72 65 64 20 66 6f | 72 6d 12 30 0d 0a 00 0d |tored fo|rm.0....|
|000044a0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000044b0| 20 20 20 20 20 11 31 2d | 36 11 33 78 20 11 31 2b |     .1-|6.3x .1+|
|000044c0| 20 39 20 3d 20 30 20 20 | 20 20 20 20 20 20 20 20 | 9 = 0  |        |
|000044d0| 20 11 32 12 31 53 69 6d | 70 6c 69 66 79 12 30 0d | .2.1Sim|plify.0.|
|000044e0| 0a 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
|000044f0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |        |       .|
|00004500| 31 33 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |13...   |        |
|00004510| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 33 |        |      .3|
|00004520| 78 20 11 31 3d 20 11 34 | 32 20 20 20 20 20 20 20 |x .1= .4|2       |
|00004530| 20 20 20 20 11 32 12 31 | 53 6f 6c 75 74 69 6f 6e |    .2.1|Solution|
|00004540| 12 30 0d 0b 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|00004550| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|00004560| 20 20 11 31 32 0d 0a 00 | 00 17 4f 0b 0d 0b 00 57 |  .12...|..O....W|
|00004570| 72 6f 6e 67 2e 20 20 57 | 65 20 63 61 6e 20 73 6f |rong.  W|e can so|
|00004580| 6c 76 65 20 74 68 65 20 | 67 69 76 65 6e 20 65 71 |lve the |given eq|
|00004590| 75 61 74 69 6f 6e 20 61 | 73 20 66 6f 6c 6c 6f 77 |uation a|s follow|
|000045a0| 73 2e 20 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |s. .0...|        |
|000045b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
|000045c0| 20 11 32 32 20 20 20 20 | 32 0d 0b 00 20 20 20 20 | .22    |2...    |
|000045d0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 11 31 |        |      .1|
|000045e0| 28 11 33 78 20 11 31 2d | 20 33 29 20 20 3d 20 11 |(.3x .1-| 3)  = .|
|000045f0| 33 78 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |3x      |     .2.|
|00004600| 31 47 69 76 65 6e 20 65 | 71 75 61 74 69 6f 6e 12 |1Given e|quation.|
|00004610| 30 0d 0a 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |0...    |        |
|00004620| 20 20 20 20 20 20 20 20 | 32 20 20 20 20 32 20 20 |        |2    2  |
|00004630| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 12 31 20 |        |     .1 |
|00004640| 20 20 20 20 20 20 20 20 | 20 32 20 20 20 20 20 20 |        | 2      |
|00004650| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 12 30 0d |        |     .0.|
|00004660| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 11 |..      |       .|
|00004670| 31 28 11 33 78 20 11 31 | 2d 20 33 29 20 20 2d 20 |1(.3x .1|- 3)  - |
|00004680| 11 33 78 20 20 11 31 3d | 20 30 20 20 20 20 20 20 |.3x  .1=| 0      |
|00004690| 20 20 20 20 20 11 32 12 | 31 53 75 62 74 72 61 63 |     .2.|1Subtrac|
|000046a0| 74 20 78 20 20 66 72 6f | 6d 20 62 6f 74 68 20 73 |t x  fro|m both s|
|000046b0| 69 64 65 73 12 30 0d 0a | 00 0d 0b 00 11 31 5b 28 |ides.0..|.....1[(|
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|000046e0| 2d 20 11 33 78 11 31 5d | 20 3d 20 30 20 20 20 20 |- .3x.1]| = 0    |
|000046f0| 20 20 20 20 20 20 20 11 | 32 12 31 46 61 63 74 6f |       .|2.1Facto|
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|00004710| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00004740| 32 12 31 53 69 6d 70 6c | 69 66 79 12 30 0d 0a 00 |2.1Simpl|ify.0...|
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|00004790| 11 31 3d 20 11 34 32 20 | 20 20 20 20 20 20 20 20 |.1= .42 |        |
|000047a0| 20 20 11 32 12 31 53 6f | 6c 75 74 69 6f 6e 12 30 |  .2.1So|lution.0|
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|000047e0| 6e 67 2e 20 20 54 68 65 | 20 67 69 76 65 6e 20 65 |ng.  The| given e|
|000047f0| 71 75 61 74 69 6f 6e 20 | 64 6f 65 73 20 68 61 76 |quation |does hav|
|00004800| 65 20 61 20 73 6f 6c 75 | 74 69 6f 6e 2e 20 20 52 |e a solu|tion.  R|
|00004810| 65 76 69 65 77 20 49 6e | 74 65 67 72 61 74 65 64 |eview In|tegrated|
|00004820| 20 0d 0a 00 45 78 61 6d | 70 6c 65 20 34 20 61 6e | ...Exam|ple 4 an|
|00004830| 64 20 74 68 65 6e 20 74 | 72 79 20 61 67 61 69 6e |d then t|ry again|
|00004840| 2e 0d 0a 00 00 17 4f 06 | 0d 0b 00 56 65 72 79 20 |......O.|...Very |
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|00004910| 20 20 20 20 20 20 20 20 | 20 20 20 11 31 3d 20 28 |        |   .1= (|
|00004920| 11 33 78 20 11 31 2d 20 | 33 29 20 20 2d 20 32 35 |.3x .1- |3)  - 25|
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|00004940| 65 63 74 2e 12 30 0d 0a | 00 20 20 20 20 20 20 11 |ect..0..|.      .|
|00004950| 32 32 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |22      |        |
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|000049c0| 20 11 31 2b 20 33 20 20 | 2d 20 31 36 20 2d 20 33 | .1+ 3  |- 16 - 3|
|000049d0| 20 20 20 20 20 20 11 32 | 12 31 41 64 64 20 61 6e |      .2|.1Add an|
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|00004a40| 20 20 20 20 20 11 32 12 | 31 47 72 6f 75 70 20 74 |     .2.|1Group t|
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|00004a90| 33 78 20 11 31 2d 20 33 | 29 20 20 2d 20 32 35 20 |3x .1- 3|)  - 25 |
|00004aa0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 32 12 |        |     .2.|
|00004ab0| 31 44 69 66 66 65 72 65 | 6e 63 65 20 6f 66 20 74 |1Differe|nce of t|
|00004ac0| 77 6f 20 73 71 75 61 72 | 65 73 12 30 0d 0a 00 00 |wo squar|es.0....|
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|00004ae0| 20 6c 6f 6f 6b 73 20 6c | 69 6b 65 20 79 6f 75 20 | looks l|ike you |
|00004af0| 61 64 64 65 64 20 39 20 | 74 6f 20 63 6f 6d 70 6c |added 9 |to compl|
|00004b00| 65 74 65 20 74 68 65 20 | 73 71 75 61 72 65 2c 20 |ete the |square, |
|00004b10| 62 75 74 20 74 68 65 6e | 20 79 6f 75 20 0d 0a 00 |but then| you ...|
|00004b20| 61 64 64 65 64 20 39 20 | 61 67 61 69 6e 20 77 68 |added 9 |again wh|
|00004b30| 65 72 65 20 79 6f 75 20 | 73 68 6f 75 6c 64 20 68 |ere you |should h|
|00004b40| 61 76 65 20 73 75 62 74 | 72 61 63 74 65 64 20 39 |ave subt|racted 9|
|00004b50| 20 69 6e 20 6f 72 64 65 | 72 20 74 6f 20 65 6e 64 | in orde|r to end|
|00004b60| 20 75 70 20 77 69 74 68 | 20 0d 0a 00 61 6e 20 65 | up with| ...an e|
|00004b70| 71 75 69 76 61 6c 65 6e | 74 20 65 78 70 72 65 73 |quivalen|t expres|
|00004b80| 73 69 6f 6e 2e 0d 0a 00 | 00 17 4f 03 0d 0b 00 53 |sion....|..O....S|
|00004b90| 6f 72 72 79 2c 20 6f 6e | 65 20 6f 66 20 74 68 65 |orry, on|e of the|
|00004ba0| 73 65 20 61 6e 73 77 65 | 72 73 20 69 73 20 63 6f |se answe|rs is co|
|00004bb0| 72 72 65 63 74 2e 20 20 | 54 72 79 20 61 67 61 69 |rrect.  |Try agai|
|00004bc0| 6e 2e 0d 0a 00 00 17 4f | 08 20 20 20 20 20 20 20 |n......O|.       |
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|00004c10| 54 6f 20 73 6f 6c 76 65 | 2c 20 77 65 20 75 73 65 |To solve|, we use|
|00004c20| 20 74 68 65 20 70 6f 73 | 69 74 69 6f 6e 20 65 71 | the pos|ition eq|
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|00004c40| 31 36 11 33 74 20 20 11 | 31 2b 20 11 33 76 20 74 |16.3t  .|1+ .3v t|
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|00004cb0| 65 20 69 6e 69 74 69 61 | 6c 20 76 65 6c 6f 63 69 |e initia|l veloci|
|00004cc0| 74 79 20 11 33 76 20 20 | 11 31 3d 20 30 2c 20 74 |ty .3v  |.1= 0, t|
|00004cd0| 68 65 20 69 6e 69 74 69 | 61 6c 20 68 65 69 67 68 |he initi|al heigh|
|00004ce0| 74 20 11 33 73 20 20 11 | 31 3d 20 31 30 35 33 2c |t .3s  .|1= 1053,|
|00004cf0| 20 61 6e 64 20 74 68 65 | 20 68 65 69 67 68 74 0d | and the| height.|
|00004d00| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |..      |        |
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|00004d60| 0b 00 11 33 73 20 11 31 | 3d 20 30 2e 20 20 53 6f |...3s .1|= 0.  So|
|00004d70| 6c 76 69 6e 67 20 74 68 | 65 20 65 71 75 61 74 69 |lving th|e equati|
|00004d80| 6f 6e 20 30 20 3d 20 2d | 31 36 11 33 74 20 20 11 |on 0 = -|16.3t  .|
|00004d90| 31 2b 20 31 30 35 33 20 | 66 6f 72 20 11 33 74 11 |1+ 1053 |for .3t.|
|00004da0| 31 2c 20 77 65 20 6f 62 | 74 61 69 6e 20 74 68 65 |1, we ob|tain the|
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|00004de0| 31 31 33 0d 0b 00 73 6f | 6c 75 74 69 6f 6e 20 11 |113...so|lution .|
|00004df0| 33 74 20 11 31 3d 20 11 | 34 32 32 32 32 32 20 7e |3t .1= .|422222 ~|
|00004e00| 20 11 31 38 2e 31 31 20 | 73 65 63 6f 6e 64 73 2e | .18.11 |seconds.|
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|00004e20| 20 20 34 0d 0a 00 00 17 | 4f 04 0d 0b 00 52 69 67 |  4.....|O....Rig|
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|00004e50| 38 2e 31 31 20 73 65 63 | 6f 6e 64 73 20 66 6f 72 |8.11 sec|onds for|
|00004e60| 20 74 68 65 20 72 6f 63 | 6b 20 74 6f 20 68 69 74 | the roc|k to hit|
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|00004ec0| 0b 00 11 31 57 72 6f 6e | 67 2e 20 20 54 6f 20 73 |...1Wron|g.  To s|
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|00005020| 73 20 11 31 3d 20 30 2e | 20 20 53 6f 6c 76 69 6e |s .1= 0.|  Solvin|
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|00005280| 33 73 20 11 31 3d 20 30 | 2e 20 20 53 6f 6c 76 69 |3s .1= 0|.  Solvi|
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|00005390| 20 55 73 69 6e 67 20 74 | 68 65 20 51 75 61 64 72 | Using t|he Quadr|
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|00005400| 11 33 63 20 11 31 3d 20 | 32 30 2c 20 77 65 20 73 |.3c .1= |20, we s|
|00005410| 6f 6c 76 65 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |olve as |follows.|
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|000054b0| 0b 00 20 20 20 20 20 20 | 20 20 20 11 31 2d 31 30 |..      |   .1-10|
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|00005510| 20 20 20 11 33 78 20 11 | 31 3d 20 11 34 32 32 32 |   .3x .|1= .4222|
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|00005530| 32 32 32 32 32 20 11 31 | 3d 20 11 34 32 32 32 32 |22222 .1|= .42222|
|00005540| 32 32 32 32 32 32 20 11 | 31 3d 20 11 34 32 32 32 |222222 .|1= .4222|
|00005550| 32 32 32 32 32 32 32 20 | 11 31 3d 20 2d 35 20 11 |2222222 |.1= -5 .|
|00005560| 34 2b 20 76 20 11 31 35 | 0d 0b 00 20 20 20 20 20 |4+ v .15|...     |
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|000055d0| 20 61 6e 64 20 11 33 63 | 20 11 31 3d 20 32 30 2c | and .3c| .1= 20,|
|000055e0| 20 77 65 20 73 6f 6c 76 | 65 20 61 73 20 66 6f 6c | we solv|e as fol|
|000055f0| 6c 6f 77 73 2e 12 30 0d | 0a 00 20 20 20 20 20 20 |lows..0.|..      |
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|00005610| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 20 20 20 |22222222|22222   |
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|00005640| 20 20 20 20 20 20 20 20 | 20 20 66 20 20 20 20 11 |        |  f    .|
|00005650| 32 32 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |22      |        |
|00005660| 20 20 20 20 20 20 20 11 | 34 66 20 20 20 20 20 20 |       .|4f      |
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|00005710| 34 32 32 32 32 32 32 32 | 32 32 32 20 11 31 3d 20 |42222222|222 .1= |
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|00005730| 20 2d 35 20 11 34 2b 20 | 76 20 11 31 35 0d 0b 00 | -5 .4+ |v .15...|
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|00005760| 20 20 20 20 20 20 20 20 | 20 32 20 20 20 20 20 20 |        | 2      |
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|00005790| 72 65 20 74 77 6f 20 64 | 69 66 66 65 72 65 6e 74 |re two d|ifferent|
|000057a0| 20 72 65 61 6c 20 73 6f | 6c 75 74 69 6f 6e 73 20 | real so|lutions |
|000057b0| 74 6f 20 74 68 69 73 20 | 65 71 75 61 74 69 6f 6e |to this |equation|
|000057c0| 20 73 69 6e 63 65 20 74 | 68 65 20 0d 0a 00 64 69 | since t|he ...di|
|000057d0| 73 63 72 69 6d 69 6e 61 | 6e 74 20 69 73 20 70 6f |scrimina|nt is po|
|000057e0| 73 69 74 69 76 65 2e 20 | 20 54 72 79 20 61 67 61 |sitive. | Try aga|
|000057f0| 69 6e 2e 0d 0a 00 00 17 | 4f 06 0d 0b 00 57 72 6f |in......|O....Wro|
|00005800| 6e 67 2e 20 20 55 73 69 | 6e 67 20 11 33 61 20 11 |ng.  Usi|ng .3a .|
|00005810| 31 3d 20 34 39 2c 20 11 | 33 62 20 11 31 3d 20 32 |1= 49, .|3b .1= 2|
|00005820| 38 2c 20 61 6e 64 20 11 | 33 63 20 11 31 3d 20 2d |8, and .|3c .1= -|
|00005830| 36 2c 20 77 65 20 73 6f | 6c 76 65 20 61 73 20 66 |6, we so|lve as f|
|00005840| 6f 6c 6c 6f 77 73 2e 12 | 30 0d 0a 00 20 20 20 20 |ollows..|0...    |
|00005850| 20 20 20 20 20 20 20 20 | 20 20 11 34 67 32 32 32 |        |  .4g222|
|00005860| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 20 20 20 |22222222|22222   |
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|00005880| 20 20 20 20 20 20 20 20 | 20 67 32 20 20 20 20 20 |        | g2     |
|00005890| 20 20 20 20 20 67 32 0d | 0b 00 20 20 20 20 20 20 |     g2.|..      |
|000058a0| 20 20 20 20 20 20 20 66 | 20 20 20 20 11 32 32 20 |       f|    .22 |
|000058b0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000058d0| 20 20 20 20 20 20 20 20 | 66 20 20 20 20 20 20 20 |        |f       |
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|00005900| 20 20 2d 20 34 28 34 39 | 29 28 2d 36 29 20 20 20 |  - 4(49|)(-6)   |
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|00005950| 11 31 3d 20 11 34 32 32 | 32 32 32 32 32 32 32 32 |.1= .422|22222222|
|00005960| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 20 |22222222|2222222 |
|00005970| 11 31 3d 20 11 34 32 32 | 32 32 32 32 32 32 32 32 |.1= .422|22222222|
|00005980| 32 32 20 11 31 3d 20 11 | 34 32 32 32 32 32 32 32 |22 .1= .|42222222|
|00005990| 32 32 32 32 32 20 11 31 | 3d 20 11 34 32 32 32 32 |22222 .1|= .42222|
|000059a0| 32 32 32 32 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |22222...|        |
|000059b0| 20 20 20 20 20 20 20 20 | 11 31 32 28 34 39 29 20 |        |.12(49) |
|000059c0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|000059e0| 39 38 20 20 20 20 20 20 | 20 20 20 20 20 20 37 0d |98      |      7.|
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|00005a10| 39 2c 20 11 33 62 20 11 | 31 3d 20 32 38 2c 20 61 |9, .3b .|1= 28, a|
|00005a20| 6e 64 20 11 33 63 20 11 | 31 3d 20 2d 36 2c 20 77 |nd .3c .|1= -6, w|
|00005a30| 65 20 73 6f 6c 76 65 20 | 61 73 20 66 6f 6c 6c 6f |e solve |as follo|
|00005a40| 77 73 2e 12 30 0d 0a 00 | 20 20 20 20 20 20 20 20 |ws..0...|        |
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|00005a60| 32 32 32 32 32 32 32 32 | 32 20 20 20 20 20 20 20 |22222222|2       |
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|00005ac0| 20 11 34 66 20 20 20 20 | 20 20 20 20 20 20 20 20 | .4f    |        |
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|00005ae0| 66 0d 0b 00 20 20 20 20 | 20 20 11 31 2d 32 38 20 |f...    |  .1-28 |
|00005af0| 11 34 2b 20 76 20 11 31 | 28 32 38 29 20 20 2d 20 |.4+ v .1|(28)  - |
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|00005b90| 32 20 11 31 3d 20 11 34 | 32 32 32 32 32 32 32 32 |2 .1= .4|22222222|
|00005ba0| 32 0d 0b 00 20 20 20 20 | 20 20 20 20 20 20 20 20 |2...    |        |
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|00005bc0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 39 38 20 |        |     98 |
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|00005c70| 20 11 31 31 30 0d 0b 00 | 52 69 67 68 74 2e 20 20 | .110...|Right.  |
|00005c80| 55 73 69 6e 67 20 74 68 | 65 20 51 75 61 64 72 61 |Using th|e Quadra|
|00005c90| 74 69 63 20 46 6f 72 6d | 75 6c 61 2c 20 77 65 20 |tic Form|ula, we |
|00005ca0| 6f 62 74 61 69 6e 20 11 | 33 78 20 11 31 3d 20 11 |obtain .|3x .1= .|
|00005cb0| 34 32 32 32 32 32 32 32 | 32 32 20 11 31 2e 0d 0b |42222222|22 .1...|
|00005cc0| 00 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |.       |        |
|00005cd0| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 20 |        |        |
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|00005cf0| 20 20 20 20 20 20 20 20 | 37 0d 0a 00 00 17 4f 04 |        |7.....O.|
|00005d00| 0d 0b 00 53 6f 72 72 79 | 2c 20 74 68 65 72 65 20 |...Sorry|, there |
|00005d10| 61 72 65 20 74 77 6f 20 | 72 65 61 6c 20 73 6f 6c |are two |real sol|
|00005d20| 75 74 69 6f 6e 73 20 74 | 6f 20 74 68 69 73 20 65 |utions t|o this e|
|00005d30| 71 75 61 74 69 6f 6e 20 | 73 69 6e 63 65 20 74 68 |quation |since th|
|00005d40| 65 20 64 69 73 63 72 69 | 6d 69 6e 61 6e 74 0d 0a |e discri|minant..|
|00005d50| 00 69 73 20 70 6f 73 69 | 74 69 76 65 2e 0d 0a 00 |.is posi|tive....|
|00005d60| 00 17 4f 06 0d 0b 00 57 | 72 6f 6e 67 2e 20 20 53 |..O....W|rong.  S|
|00005d70| 69 6e 63 65 20 11 33 61 | 20 11 31 3d 20 33 2c 20 |ince .3a| .1= 3, |
|00005d80| 11 33 62 20 11 31 3d 20 | 34 2c 20 61 6e 64 20 11 |.3b .1= |4, and .|
|00005d90| 33 63 20 11 31 3d 20 31 | 2c 20 74 68 65 20 64 69 |3c .1= 1|, the di|
|00005da0| 73 63 72 69 6d 69 6e 61 | 6e 74 20 69 73 20 0d 0a |scrimina|nt is ..|
|00005db0| 00 20 20 20 20 20 20 11 | 32 32 20 20 20 20 20 20 |.      .|22      |
|00005dc0| 20 20 20 20 32 0d 0b 00 | 20 20 20 20 20 11 33 62 |    2...|     .3b|
|00005dd0| 20 20 11 31 2d 20 34 11 | 33 61 63 20 11 31 3d 20 |  .1- 4.|3ac .1= |
|00005de0| 34 20 20 2d 20 34 28 33 | 29 28 31 29 20 3d 20 31 |4  - 4(3|)(1) = 1|
|00005df0| 36 20 2d 20 31 32 20 3d | 20 34 2e 0d 0a 00 0d 0b |6 - 12 =| 4......|
|00005e00| 00 54 68 65 72 65 66 6f | 72 65 2c 20 74 68 65 72 |.Therefo|re, ther|
|00005e10| 65 20 61 72 65 20 74 77 | 6f 20 64 69 66 66 65 72 |e are tw|o differ|
|00005e20| 65 6e 74 20 72 65 61 6c | 20 73 6f 6c 75 74 69 6f |ent real| solutio|
|00005e30| 6e 73 2e 0d 0a 00 00 17 | 4f 04 0d 0b 00 56 65 72 |ns......|O....Ver|
|00005e40| 79 20 67 6f 6f 64 2e 20 | 20 53 69 6e 63 65 20 74 |y good. | Since t|
|00005e50| 68 65 20 64 69 73 63 72 | 69 6d 69 6e 61 6e 74 20 |he discr|iminant |
|00005e60| 69 73 20 70 6f 73 69 74 | 69 76 65 2c 20 74 68 65 |is posit|ive, the|
|00005e70| 72 65 20 61 72 65 20 74 | 77 6f 20 64 69 66 66 65 |re are t|wo diffe|
|00005e80| 72 65 6e 74 0d 0a 00 72 | 65 61 6c 20 73 6f 6c 75 |rent...r|eal solu|
|00005e90| 74 69 6f 6e 73 20 74 6f | 20 74 68 69 73 20 65 71 |tions to| this eq|
|00005ea0| 75 61 74 69 6f 6e 2e 0d | 0a 00 00 17 4f 04 0d 0b |uation..|....O...|
|00005eb0| 00 57 72 6f 6e 67 2e 20 | 20 54 68 65 72 65 20 69 |.Wrong. | There i|
|00005ec0| 73 20 6e 6f 20 70 6f 73 | 73 69 62 6c 65 20 77 61 |s no pos|sible wa|
|00005ed0| 79 20 61 6e 79 20 71 75 | 61 64 72 61 74 69 63 20 |y any qu|adratic |
|00005ee0| 65 71 75 61 74 69 6f 6e | 20 63 61 6e 20 68 61 76 |equation| can hav|
|00005ef0| 65 20 6d 6f 72 65 20 74 | 68 61 6e 20 0d 0a 00 74 |e more t|han ...t|
|00005f00| 77 6f 20 72 65 61 6c 20 | 73 6f 6c 75 74 69 6f 6e |wo real |solution|
|00005f10| 73 2e 20 20 52 65 76 69 | 65 77 20 47 75 69 64 65 |s.  Revi|ew Guide|
|00005f20| 64 20 45 78 61 6d 70 6c | 65 20 38 20 61 6e 64 20 |d Exampl|e 8 and |
|00005f30| 74 68 65 6e 20 74 72 79 | 20 61 67 61 69 6e 2e 0d |then try| again..|
|00005f40| 0a 00 00 17 4f 06 0d 0b | 00 57 72 6f 6e 67 2e 20 |....O...|.Wrong. |
|00005f50| 20 53 69 6e 63 65 20 11 | 33 61 20 11 31 3d 20 33 | Since .|3a .1= 3|
|00005f60| 2c 20 11 33 62 20 11 31 | 3d 20 34 2c 20 61 6e 64 |, .3b .1|= 4, and|
|00005f70| 20 11 33 63 20 11 31 3d | 20 31 2c 20 74 68 65 20 | .3c .1=| 1, the |
|00005f80| 64 69 73 63 72 69 6d 69 | 6e 61 6e 74 20 69 73 20 |discrimi|nant is |
|00005f90| 0d 0a 00 20 20 20 20 20 | 20 11 32 32 20 20 20 20 |...     | .22    |
|00005fa0| 20 20 20 20 20 20 32 0d | 0b 00 20 20 20 20 20 11 |      2.|..     .|
|00005fb0| 33 62 20 20 11 31 2d 20 | 34 11 33 61 63 20 11 31 |3b  .1- |4.3ac .1|
|00005fc0| 3d 20 34 20 20 2d 20 34 | 28 33 29 28 31 29 20 3d |= 4  - 4|(3)(1) =|
|00005fd0| 20 31 36 20 2d 20 31 32 | 20 3d 20 34 2e 0d 0a 00 | 16 - 12| = 4....|
|00005fe0| 0d 0b 00 54 68 65 72 65 | 66 6f 72 65 2c 20 74 68 |...There|fore, th|
|00005ff0| 65 72 65 20 61 72 65 20 | 74 77 6f 20 64 69 66 66 |ere are |two diff|
|00006000| 65 72 65 6e 74 20 72 65 | 61 6c 20 73 6f 6c 75 74 |erent re|al solut|
|00006010| 69 6f 6e 73 2e 0d 0a 00 | 00 17 4f 03 0d 0b 00 43 |ions....|..O....C|
|00006020| 6f 72 72 65 63 74 2e 20 | 20 43 68 65 63 6b 20 74 |orrect. | Check t|
|00006030| 68 65 73 65 20 74 77 6f | 20 73 6f 6c 75 74 69 6f |hese two| solutio|
|00006040| 6e 73 20 69 6e 20 74 68 | 65 20 6f 72 69 67 69 6e |ns in th|e origin|
|00006050| 61 6c 20 65 71 75 61 74 | 69 6f 6e 2e 0d 0a 00 00 |al equat|ion.....|
|00006060| 17 4f 06 0d 0b 00 57 72 | 6f 6e 67 2e 20 20 55 73 |.O....Wr|ong.  Us|
|00006070| 69 6e 67 20 11 33 61 20 | 11 31 3d 20 32 35 2c 20 |ing .3a |.1= 25, |
|00006080| 11 33 62 20 11 31 3d 20 | 37 30 2c 20 61 6e 64 20 |.3b .1= |70, and |
|00006090| 11 33 63 20 11 31 3d 20 | 34 34 2c 20 77 65 20 73 |.3c .1= |44, we s|
|000060a0| 6f 6c 76 65 20 61 73 20 | 66 6f 6c 6c 6f 77 73 2e |olve as |follows.|
|000060b0| 12 30 0d 0a 00 20 20 20 | 20 20 20 20 20 20 20 20 |.0...   |        |
|000060c0| 20 20 20 11 34 67 32 32 | 32 32 32 32 32 32 32 32 |   .4g22|22222222|
|000060d0| 32 32 32 32 32 32 20 20 | 20 20 20 20 20 20 20 20 |222222  |        |
|000060e0| 20 67 32 32 20 20 20 20 | 20 20 20 20 20 20 20 20 | g22    |        |
|000060f0| 20 67 20 20 20 20 20 20 | 20 20 20 20 67 0d 0b 00 | g      |    g...|
|00006100| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 66 20 20 |        |     f  |
|00006110| 20 20 11 32 32 20 20 20 | 20 20 20 20 20 20 20 20 |  .22   |        |
|00006120| 20 20 20 20 20 20 20 20 | 20 20 20 11 34 66 20 20 |        |   .4f  |
|00006130| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 66 20 20 |        |     f  |
|00006140| 20 20 20 20 20 20 20 20 | 66 0d 0b 00 20 20 20 20 |        |f...    |
|00006150| 20 20 11 31 2d 37 30 20 | 11 34 2b 20 76 20 11 31 |  .1-70 |.4+ v .1|
|00006160| 28 37 30 29 20 20 2d 20 | 34 28 32 35 29 28 34 34 |(70)  - |4(25)(44|
|00006170| 29 20 20 20 2d 37 30 20 | 11 34 2b 20 76 20 11 31 |)   -70 |.4+ v .1|
|00006180| 35 30 30 20 20 20 2d 37 | 30 20 11 34 2b 20 11 31 |500   -7|0 .4+ .1|
|00006190| 31 30 11 34 76 20 11 31 | 35 20 20 20 20 37 20 20 |10.4v .1|5    7  |
|000061a0| 20 11 34 76 20 11 31 35 | 0d 0b 00 20 20 11 33 68 | .4v .15|...  .3h|
|000061b0| 20 11 31 3d 20 11 34 32 | 32 32 32 32 32 32 32 32 | .1= .42|22222222|
|000061c0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|000061d0| 20 11 31 3d 20 11 34 32 | 32 32 32 32 32 32 32 32 | .1= .42|22222222|
|000061e0| 32 32 20 11 31 3d 20 11 | 34 32 32 32 32 32 32 32 |22 .1= .|42222222|
|000061f0| 32 32 32 32 20 11 31 3d | 20 2d 11 34 32 20 2b 20 |2222 .1=| -.42 + |
|00006200| 32 32 32 0d 0b 00 20 20 | 20 20 20 20 20 20 20 20 |222...  |        |
|00006210| 20 20 20 20 20 20 11 31 | 32 28 32 35 29 20 20 20 |      .1|2(25)   |
|00006220| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 35 30 |        |      50|
|00006230| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 35 30 20 |        |     50 |
|00006240| 20 20 20 20 20 20 20 35 | 20 20 20 20 20 35 0d 0a |       5|     5..|
|00006250| 00 00 17 4f 06 0d 0b 00 | 57 72 6f 6e 67 2e 20 20 |...O....|Wrong.  |
|00006260| 55 73 69 6e 67 20 11 33 | 61 20 11 31 3d 20 32 35 |Using .3|a .1= 25|
|00006270| 2c 20 11 33 62 20 11 31 | 3d 20 37 30 2c 20 61 6e |, .3b .1|= 70, an|
|00006280| 64 20 11 33 63 20 11 31 | 3d 20 34 34 2c 20 77 65 |d .3c .1|= 44, we|
|00006290| 20 73 6f 6c 76 65 20 61 | 73 20 66 6f 6c 6c 6f 77 | solve a|s follow|
|000062a0| 73 2e 12 30 0d 0a 00 20 | 20 20 20 20 20 20 20 20 |s..0... |        |
|000062b0| 20 20 20 20 20 11 34 67 | 32 32 32 32 32 32 32 32 |     .4g|22222222|
|000062c0| 32 32 32 32 32 32 32 32 | 20 20 20 20 20 20 20 20 |22222222|        |
|000062d0| 20 20 20 67 32 32 20 20 | 20 20 20 20 20 20 20 20 |   g22  |        |
|000062e0| 20 20 20 67 20 20 20 20 | 20 20 20 20 20 20 67 0d |   g    |      g.|
|000062f0| 0b 00 20 20 20 20 20 20 | 20 20 20 20 20 20 20 66 |..      |       f|
|00006300| 20 20 20 20 11 32 32 20 | 20 20 20 20 20 20 20 20 |    .22 |        |
|00006310| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 11 34 66 |        |     .4f|
|00006320| 20 20 20 20 20 20 20 20 | 20 20 20 20 20 20 20 66 |        |       f|
|00006330| 20 20 20 20 20 20 20 20 | 20 20 66 0d 0b 00 20 20 |        |  f...  |
|00006340| 20 20 20 20 11 31 2d 37 | 30 20 11 34 2b 20 76 20 |    .1-7|0 .4+ v |
|00006350| 11 31 28 37 30 29 20 20 | 2d 20 34 28 32 35 29 28 |.1(70)  |- 4(25)(|
|00006360| 34 34 29 20 20 20 2d 37 | 30 20 11 34 2b 20 76 20 |44)   -7|0 .4+ v |
|00006370| 11 31 35 30 30 20 20 20 | 2d 37 30 20 11 34 2b 20 |.1500   |-70 .4+ |
|00006380| 11 31 31 30 11 34 76 20 | 11 31 35 20 20 20 20 37 |.110.4v |.15    7|
|00006390| 20 20 20 11 34 76 20 11 | 31 35 0d 0b 00 20 20 11 |   .4v .|15...  .|
|000063a0| 33 68 20 11 31 3d 20 11 | 34 32 32 32 32 32 32 32 |3h .1= .|42222222|
|000063b0| 32 32 32 32 32 32 32 32 | 32 32 32 32 32 32 32 32 |22222222|22222222|
|000063c0| 32 32 20 11 31 3d 20 11 | 34 32 32 32 32 32 32 32 |22 .1= .|42222222|
|000063d0| 32 32 32 32 20 11 31 3d | 20 11 34 32 32 32 32 32 |2222 .1=| .422222|
|000063e0| 32 32 32 32 32 32 20 11 | 31 3d 20 2d 11 34 32 20 |222222 .|1= -.42 |
|000063f0| 2b 20 32 32 32 0d 0b 00 | 20 20 20 20 20 20 20 20 |+ 222...|        |
+--------+-------------------------+-------------------------+--------+--------+
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