(x-4)(x-5)=10

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


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

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

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

Multiply (-4 + x) * (-5 + x)
(-4(-5 + x) + x(-5 + x)) = 10
((-5 * -4 + x * -4) + x(-5 + x)) = 10
((20 + -4x) + x(-5 + x)) = 10
(20 + -4x + (-5 * x + x * x)) = 10
(20 + -4x + (-5x + x2)) = 10

Combine like terms: -4x + -5x = -9x
(20 + -9x + x2) = 10

Solving
20 + -9x + x2 = 10

Solving for variable 'x'.

Reorder the terms:
20 + -10 + -9x + x2 = 10 + -10

Combine like terms: 20 + -10 = 10
10 + -9x + x2 = 10 + -10

Combine like terms: 10 + -10 = 0
10 + -9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + -9x + -10 + x2 = 0 + -10

Reorder the terms:
10 + -10 + -9x + x2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -9x + x2 = 0 + -10
-9x + x2 = 0 + -10

Combine like terms: 0 + -10 = -10
-9x + x2 = -10

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = -10 + 20.25

Reorder the terms:
20.25 + -9x + x2 = -10 + 20.25

Combine like terms: -10 + 20.25 = 10.25
20.25 + -9x + x2 = 10.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(x + -4.5) equal to 3.201562119 and -3.201562119.


Subproblem 1
x + -4.5 = 3.201562119

Simplifying
x + -4.5 = 3.201562119

Reorder the terms:
-4.5 + x = 3.201562119

Solving
-4.5 + x = 3.201562119

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = 3.201562119 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 3.201562119 + 4.5
x = 3.201562119 + 4.5

Combine like terms: 3.201562119 + 4.5 = 7.701562119
x = 7.701562119

Simplifying
x = 7.701562119


Subproblem 2
x + -4.5 = -3.201562119

Simplifying
x + -4.5 = -3.201562119

Reorder the terms:
-4.5 + x = -3.201562119

Solving
-4.5 + x = -3.201562119

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = -3.201562119 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -3.201562119 + 4.5
x = -3.201562119 + 4.5

Combine like terms: -3.201562119 + 4.5 = 1.298437881
x = 1.298437881

Simplifying
x = 1.298437881


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.701562119, 1.298437881}

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