(2x-3)(2x-3)-135=0

Solution for (2x-3)(2x-3)-135=0 equation:

Simplifying
(2x + -3)(2x + -3) + -135 = 0

Reorder the terms:
(-3 + 2x)(2x + -3) + -135 = 0

Reorder the terms:
(-3 + 2x)(-3 + 2x) + -135 = 0

Multiply (-3 + 2x) * (-3 + 2x)
(-3(-3 + 2x) + 2x * (-3 + 2x)) + -135 = 0
((-3 * -3 + 2x * -3) + 2x * (-3 + 2x)) + -135 = 0
((9 + -6x) + 2x * (-3 + 2x)) + -135 = 0
(9 + -6x + (-3 * 2x + 2x * 2x)) + -135 = 0
(9 + -6x + (-6x + 4x2)) + -135 = 0

Combine like terms: -6x + -6x = -12x
(9 + -12x + 4x2) + -135 = 0

Reorder the terms:
9 + -135 + -12x + 4x2 = 0

Combine like terms: 9 + -135 = -126
-126 + -12x + 4x2 = 0

Solving
-126 + -12x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-63 + -6x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-63 + -6x + 2x2)' equal to zero and attempt to solve:

Simplifying
-63 + -6x + 2x2 = 0

Solving
-63 + -6x + 2x2 = 0

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

Divide each side by '2'.
-31.5 + -3x + x2 = 0

Move the constant term to the right:

Add '31.5' to each side of the equation.
-31.5 + -3x + 31.5 + x2 = 0 + 31.5

Reorder the terms:
-31.5 + 31.5 + -3x + x2 = 0 + 31.5

Combine like terms: -31.5 + 31.5 = 0.0
0.0 + -3x + x2 = 0 + 31.5
-3x + x2 = 0 + 31.5

Combine like terms: 0 + 31.5 = 31.5
-3x + x2 = 31.5

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 31.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 31.5 + 2.25

Combine like terms: 31.5 + 2.25 = 33.75
2.25 + -3x + x2 = 33.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 33.75

Calculate the square root of the right side: 5.809475019

Break this problem into two subproblems by setting 
(x + -1.5) equal to 5.809475019 and -5.809475019.

Subproblem 1
x + -1.5 = 5.809475019

Simplifying
x + -1.5 = 5.809475019

Reorder the terms:
-1.5 + x = 5.809475019

Solving
-1.5 + x = 5.809475019

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 5.809475019 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 5.809475019 + 1.5
x = 5.809475019 + 1.5

Combine like terms: 5.809475019 + 1.5 = 7.309475019
x = 7.309475019

Simplifying
x = 7.309475019

Subproblem 2
x + -1.5 = -5.809475019

Simplifying
x + -1.5 = -5.809475019

Reorder the terms:
-1.5 + x = -5.809475019

Solving
-1.5 + x = -5.809475019

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -5.809475019 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -5.809475019 + 1.5
x = -5.809475019 + 1.5

Combine like terms: -5.809475019 + 1.5 = -4.309475019
x = -4.309475019

Simplifying
x = -4.309475019

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.309475019, -4.309475019}
Solution
x = {7.309475019, -4.309475019}