(2x-3)(2x-3)=5

Solution for (2x-3)(2x-3)=5 equation:

Simplifying
(2x + -3)(2x + -3) = 5

Reorder the terms:
(-3 + 2x)(2x + -3) = 5

Reorder the terms:
(-3 + 2x)(-3 + 2x) = 5

Multiply (-3 + 2x) * (-3 + 2x)
(-3(-3 + 2x) + 2x * (-3 + 2x)) = 5
((-3 * -3 + 2x * -3) + 2x * (-3 + 2x)) = 5
((9 + -6x) + 2x * (-3 + 2x)) = 5
(9 + -6x + (-3 * 2x + 2x * 2x)) = 5
(9 + -6x + (-6x + 4x2)) = 5

Combine like terms: -6x + -6x = -12x
(9 + -12x + 4x2) = 5

Solving
9 + -12x + 4x2 = 5

Solving for variable 'x'.

Reorder the terms:
9 + -5 + -12x + 4x2 = 5 + -5

Combine like terms: 9 + -5 = 4
4 + -12x + 4x2 = 5 + -5

Combine like terms: 5 + -5 = 0
4 + -12x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(1 + -3x + x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(1 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
1 + -3x + x2 = 0

Solving
1 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -3x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3x + x2 = 0 + -1
-3x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3x + x2 = -1

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 = -1 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3x + x2 = 1.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x + -1.5 = 1.118033989

Simplifying
x + -1.5 = 1.118033989

Reorder the terms:
-1.5 + x = 1.118033989

Solving
-1.5 + x = 1.118033989

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 = 1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.118033989 + 1.5
x = 1.118033989 + 1.5

Combine like terms: 1.118033989 + 1.5 = 2.618033989
x = 2.618033989

Simplifying
x = 2.618033989

Subproblem 2
x + -1.5 = -1.118033989

Simplifying
x + -1.5 = -1.118033989

Reorder the terms:
-1.5 + x = -1.118033989

Solving
-1.5 + x = -1.118033989

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 = -1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.118033989 + 1.5
x = -1.118033989 + 1.5

Combine like terms: -1.118033989 + 1.5 = 0.381966011
x = 0.381966011

Simplifying
x = 0.381966011

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.618033989, 0.381966011}
Solution
x = {2.618033989, 0.381966011}