12(x-6)(x+9)-5(x+7)(x-8)=32(x-5)(x-10)

Solution for 12(x-6)(x+9)-5(x+7)(x-8)=32(x-5)(x-10) equation:

Simplifying
12(x + -6)(x + 9) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)

Reorder the terms:
12(-6 + x)(x + 9) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)

Reorder the terms:
12(-6 + x)(9 + x) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)

Multiply (-6 + x) * (9 + x)
12(-6(9 + x) + x(9 + x)) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
12((9 * -6 + x * -6) + x(9 + x)) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
12((-54 + -6x) + x(9 + x)) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
12(-54 + -6x + (9 * x + x * x)) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
12(-54 + -6x + (9x + x2)) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)

Combine like terms: -6x + 9x = 3x
12(-54 + 3x + x2) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
(-54 * 12 + 3x * 12 + x2 * 12) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)
(-648 + 36x + 12x2) + -5(x + 7)(x + -8) = 32(x + -5)(x + -10)

Reorder the terms:
-648 + 36x + 12x2 + -5(7 + x)(x + -8) = 32(x + -5)(x + -10)

Reorder the terms:
-648 + 36x + 12x2 + -5(7 + x)(-8 + x) = 32(x + -5)(x + -10)

Multiply (7 + x) * (-8 + x)
-648 + 36x + 12x2 + -5(7(-8 + x) + x(-8 + x)) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + -5((-8 * 7 + x * 7) + x(-8 + x)) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + -5((-56 + 7x) + x(-8 + x)) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + -5(-56 + 7x + (-8 * x + x * x)) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + -5(-56 + 7x + (-8x + x2)) = 32(x + -5)(x + -10)

Combine like terms: 7x + -8x = -1x
-648 + 36x + 12x2 + -5(-56 + -1x + x2) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + (-56 * -5 + -1x * -5 + x2 * -5) = 32(x + -5)(x + -10)
-648 + 36x + 12x2 + (280 + 5x + -5x2) = 32(x + -5)(x + -10)

Reorder the terms:
-648 + 280 + 36x + 5x + 12x2 + -5x2 = 32(x + -5)(x + -10)

Combine like terms: -648 + 280 = -368
-368 + 36x + 5x + 12x2 + -5x2 = 32(x + -5)(x + -10)

Combine like terms: 36x + 5x = 41x
-368 + 41x + 12x2 + -5x2 = 32(x + -5)(x + -10)

Combine like terms: 12x2 + -5x2 = 7x2
-368 + 41x + 7x2 = 32(x + -5)(x + -10)

Reorder the terms:
-368 + 41x + 7x2 = 32(-5 + x)(x + -10)

Reorder the terms:
-368 + 41x + 7x2 = 32(-5 + x)(-10 + x)

Multiply (-5 + x) * (-10 + x)
-368 + 41x + 7x2 = 32(-5(-10 + x) + x(-10 + x))
-368 + 41x + 7x2 = 32((-10 * -5 + x * -5) + x(-10 + x))
-368 + 41x + 7x2 = 32((50 + -5x) + x(-10 + x))
-368 + 41x + 7x2 = 32(50 + -5x + (-10 * x + x * x))
-368 + 41x + 7x2 = 32(50 + -5x + (-10x + x2))

Combine like terms: -5x + -10x = -15x
-368 + 41x + 7x2 = 32(50 + -15x + x2)
-368 + 41x + 7x2 = (50 * 32 + -15x * 32 + x2 * 32)
-368 + 41x + 7x2 = (1600 + -480x + 32x2)

Solving
-368 + 41x + 7x2 = 1600 + -480x + 32x2

Solving for variable 'x'.

Reorder the terms:
-368 + -1600 + 41x + 480x + 7x2 + -32x2 = 1600 + -480x + 32x2 + -1600 + 480x + -32x2

Combine like terms: -368 + -1600 = -1968
-1968 + 41x + 480x + 7x2 + -32x2 = 1600 + -480x + 32x2 + -1600 + 480x + -32x2

Combine like terms: 41x + 480x = 521x
-1968 + 521x + 7x2 + -32x2 = 1600 + -480x + 32x2 + -1600 + 480x + -32x2

Combine like terms: 7x2 + -32x2 = -25x2
-1968 + 521x + -25x2 = 1600 + -480x + 32x2 + -1600 + 480x + -32x2

Reorder the terms:
-1968 + 521x + -25x2 = 1600 + -1600 + -480x + 480x + 32x2 + -32x2

Combine like terms: 1600 + -1600 = 0
-1968 + 521x + -25x2 = 0 + -480x + 480x + 32x2 + -32x2
-1968 + 521x + -25x2 = -480x + 480x + 32x2 + -32x2

Combine like terms: -480x + 480x = 0
-1968 + 521x + -25x2 = 0 + 32x2 + -32x2
-1968 + 521x + -25x2 = 32x2 + -32x2

Combine like terms: 32x2 + -32x2 = 0
-1968 + 521x + -25x2 = 0

Begin completing the square.  Divide all terms by
-25 the coefficient of the squared term: 

Divide each side by '-25'.
78.72 + -20.84x + x2 = 0

Move the constant term to the right:

Add '-78.72' to each side of the equation.
78.72 + -20.84x + -78.72 + x2 = 0 + -78.72

Reorder the terms:
78.72 + -78.72 + -20.84x + x2 = 0 + -78.72

Combine like terms: 78.72 + -78.72 = 0.00
0.00 + -20.84x + x2 = 0 + -78.72
-20.84x + x2 = 0 + -78.72

Combine like terms: 0 + -78.72 = -78.72
-20.84x + x2 = -78.72

The x term is -20.84x.  Take half its coefficient (-10.42).
Square it (108.5764) and add it to both sides.

Add '108.5764' to each side of the equation.
-20.84x + 108.5764 + x2 = -78.72 + 108.5764

Reorder the terms:
108.5764 + -20.84x + x2 = -78.72 + 108.5764

Combine like terms: -78.72 + 108.5764 = 29.8564
108.5764 + -20.84x + x2 = 29.8564

Factor a perfect square on the left side:
(x + -10.42)(x + -10.42) = 29.8564

Calculate the square root of the right side: 5.464101024

Break this problem into two subproblems by setting 
(x + -10.42) equal to 5.464101024 and -5.464101024.

Subproblem 1
x + -10.42 = 5.464101024

Simplifying
x + -10.42 = 5.464101024

Reorder the terms:
-10.42 + x = 5.464101024

Solving
-10.42 + x = 5.464101024

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.42' to each side of the equation.
-10.42 + 10.42 + x = 5.464101024 + 10.42

Combine like terms: -10.42 + 10.42 = 0.00
0.00 + x = 5.464101024 + 10.42
x = 5.464101024 + 10.42

Combine like terms: 5.464101024 + 10.42 = 15.884101024
x = 15.884101024

Simplifying
x = 15.884101024

Subproblem 2
x + -10.42 = -5.464101024

Simplifying
x + -10.42 = -5.464101024

Reorder the terms:
-10.42 + x = -5.464101024

Solving
-10.42 + x = -5.464101024

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.42' to each side of the equation.
-10.42 + 10.42 + x = -5.464101024 + 10.42

Combine like terms: -10.42 + 10.42 = 0.00
0.00 + x = -5.464101024 + 10.42
x = -5.464101024 + 10.42

Combine like terms: -5.464101024 + 10.42 = 4.955898976
x = 4.955898976

Simplifying
x = 4.955898976

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {15.884101024, 4.955898976}