(x-6)(x-5)+(x-7)(x-4)=10

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Solution for (x-6)(x-5)+(x-7)(x-4)=10 equation: 


Simplifying
(x + -6)(x + -5) + (x + -7)(x + -4) = 10

Reorder the terms:
(-6 + x)(x + -5) + (x + -7)(x + -4) = 10

Reorder the terms:
(-6 + x)(-5 + x) + (x + -7)(x + -4) = 10

Multiply (-6 + x) * (-5 + x)
(-6(-5 + x) + x(-5 + x)) + (x + -7)(x + -4) = 10
((-5 * -6 + x * -6) + x(-5 + x)) + (x + -7)(x + -4) = 10
((30 + -6x) + x(-5 + x)) + (x + -7)(x + -4) = 10
(30 + -6x + (-5 * x + x * x)) + (x + -7)(x + -4) = 10
(30 + -6x + (-5x + x2)) + (x + -7)(x + -4) = 10

Combine like terms: -6x + -5x = -11x
(30 + -11x + x2) + (x + -7)(x + -4) = 10

Reorder the terms:
30 + -11x + x2 + (-7 + x)(x + -4) = 10

Reorder the terms:
30 + -11x + x2 + (-7 + x)(-4 + x) = 10

Multiply (-7 + x) * (-4 + x)
30 + -11x + x2 + (-7(-4 + x) + x(-4 + x)) = 10
30 + -11x + x2 + ((-4 * -7 + x * -7) + x(-4 + x)) = 10
30 + -11x + x2 + ((28 + -7x) + x(-4 + x)) = 10
30 + -11x + x2 + (28 + -7x + (-4 * x + x * x)) = 10
30 + -11x + x2 + (28 + -7x + (-4x + x2)) = 10

Combine like terms: -7x + -4x = -11x
30 + -11x + x2 + (28 + -11x + x2) = 10

Reorder the terms:
30 + 28 + -11x + -11x + x2 + x2 = 10

Combine like terms: 30 + 28 = 58
58 + -11x + -11x + x2 + x2 = 10

Combine like terms: -11x + -11x = -22x
58 + -22x + x2 + x2 = 10

Combine like terms: x2 + x2 = 2x2
58 + -22x + 2x2 = 10

Solving
58 + -22x + 2x2 = 10

Solving for variable 'x'.

Reorder the terms:
58 + -10 + -22x + 2x2 = 10 + -10

Combine like terms: 58 + -10 = 48
48 + -22x + 2x2 = 10 + -10

Combine like terms: 10 + -10 = 0
48 + -22x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(24 + -11x + x2) = 0

Factor a trinomial.
2((3 + -1x)(8 + -1x)) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(3 + -1x)' equal to zero and attempt to solve:

Simplifying
3 + -1x = 0

Solving
3 + -1x = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1x = 0 + -3
-1x = 0 + -3

Combine like terms: 0 + -3 = -3
-1x = -3

Divide each side by '-1'.
x = 3

Simplifying
x = 3

Subproblem 2
Set the factor '(8 + -1x)' equal to zero and attempt to solve:

Simplifying
8 + -1x = 0

Solving
8 + -1x = 0

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

Add '-8' to each side of the equation.
8 + -8 + -1x = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -1x = 0 + -8
-1x = 0 + -8

Combine like terms: 0 + -8 = -8
-1x = -8

Divide each side by '-1'.
x = 8

Simplifying
x = 8
Solution
x = {3, 8}

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