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

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 = 3x
30 + -11x + x2 + (-28 + 3x + x2) = 10

Reorder the terms:
30 + -28 + -11x + 3x + x2 + x2 = 10

Combine like terms: 30 + -28 = 2
2 + -11x + 3x + x2 + x2 = 10

Combine like terms: -11x + 3x = -8x
2 + -8x + x2 + x2 = 10

Combine like terms: x2 + x2 = 2x2
2 + -8x + 2x2 = 10

Solving
2 + -8x + 2x2 = 10

Solving for variable 'x'.

Reorder the terms:
2 + -10 + -8x + 2x2 = 10 + -10

Combine like terms: 2 + -10 = -8
-8 + -8x + 2x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-8 + -8x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + -4x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-4 + -4x + x2)' equal to zero and attempt to solve:

Simplifying
-4 + -4x + x2 = 0

Solving
-4 + -4x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -4x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -4x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -4x + x2 = 0 + 4
-4x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-4x + x2 = 4

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = 4 + 4

Reorder the terms:
4 + -4x + x2 = 4 + 4

Combine like terms: 4 + 4 = 8
4 + -4x + x2 = 8

Factor a perfect square on the left side:
(x + -2)(x + -2) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(x + -2) equal to 2.828427125 and -2.828427125.

Subproblem 1
x + -2 = 2.828427125

Simplifying
x + -2 = 2.828427125

Reorder the terms:
-2 + x = 2.828427125

Solving
-2 + x = 2.828427125

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = 2.828427125 + 2

Combine like terms: -2 + 2 = 0
0 + x = 2.828427125 + 2
x = 2.828427125 + 2

Combine like terms: 2.828427125 + 2 = 4.828427125
x = 4.828427125

Simplifying
x = 4.828427125

Subproblem 2
x + -2 = -2.828427125

Simplifying
x + -2 = -2.828427125

Reorder the terms:
-2 + x = -2.828427125

Solving
-2 + x = -2.828427125

Solving for variable 'x'.

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

Add '2' to each side of the equation.
-2 + 2 + x = -2.828427125 + 2

Combine like terms: -2 + 2 = 0
0 + x = -2.828427125 + 2
x = -2.828427125 + 2

Combine like terms: -2.828427125 + 2 = -0.828427125
x = -0.828427125

Simplifying
x = -0.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.828427125, -0.828427125}
Solution
x = {4.828427125, -0.828427125}