(x-5)(x-3)-x=5

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Solution for (x-5)(x-3)-x=5 equation: 


Simplifying
(x + -5)(x + -3) + -1x = 5

Reorder the terms:
(-5 + x)(x + -3) + -1x = 5

Reorder the terms:
(-5 + x)(-3 + x) + -1x = 5

Multiply (-5 + x) * (-3 + x)
(-5(-3 + x) + x(-3 + x)) + -1x = 5
((-3 * -5 + x * -5) + x(-3 + x)) + -1x = 5
((15 + -5x) + x(-3 + x)) + -1x = 5
(15 + -5x + (-3 * x + x * x)) + -1x = 5
(15 + -5x + (-3x + x2)) + -1x = 5

Combine like terms: -5x + -3x = -8x
(15 + -8x + x2) + -1x = 5

Reorder the terms:
15 + -8x + -1x + x2 = 5

Combine like terms: -8x + -1x = -9x
15 + -9x + x2 = 5

Solving
15 + -9x + x2 = 5

Solving for variable 'x'.

Reorder the terms:
15 + -5 + -9x + x2 = 5 + -5

Combine like terms: 15 + -5 = 10
10 + -9x + x2 = 5 + -5

Combine like terms: 5 + -5 = 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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