(2x-3)(2x-3)=7

Solution for (2x-3)(2x-3)=7 equation:

Simplifying
(2x + -3)(2x + -3) = 7

Reorder the terms:
(-3 + 2x)(2x + -3) = 7

Reorder the terms:
(-3 + 2x)(-3 + 2x) = 7

Multiply (-3 + 2x) * (-3 + 2x)
(-3(-3 + 2x) + 2x * (-3 + 2x)) = 7
((-3 * -3 + 2x * -3) + 2x * (-3 + 2x)) = 7
((9 + -6x) + 2x * (-3 + 2x)) = 7
(9 + -6x + (-3 * 2x + 2x * 2x)) = 7
(9 + -6x + (-6x + 4x2)) = 7

Combine like terms: -6x + -6x = -12x
(9 + -12x + 4x2) = 7

Solving
9 + -12x + 4x2 = 7

Solving for variable 'x'.

Reorder the terms:
9 + -7 + -12x + 4x2 = 7 + -7

Combine like terms: 9 + -7 = 2
2 + -12x + 4x2 = 7 + -7

Combine like terms: 7 + -7 = 0
2 + -12x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + -6x + 2x2) = 0

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

Simplifying
1 + -6x + 2x2 = 0

Solving
1 + -6x + 2x2 = 0

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

Divide each side by '2'.
0.5 + -3x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3x + x2 = 0 + -0.5
-3x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3x + x2 = -0.5

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 = -0.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -0.5 + 2.25

Combine like terms: -0.5 + 2.25 = 1.75
2.25 + -3x + x2 = 1.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
x + -1.5 = 1.322875656

Simplifying
x + -1.5 = 1.322875656

Reorder the terms:
-1.5 + x = 1.322875656

Solving
-1.5 + x = 1.322875656

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 = 1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.322875656 + 1.5
x = 1.322875656 + 1.5

Combine like terms: 1.322875656 + 1.5 = 2.822875656
x = 2.822875656

Simplifying
x = 2.822875656

Subproblem 2
x + -1.5 = -1.322875656

Simplifying
x + -1.5 = -1.322875656

Reorder the terms:
-1.5 + x = -1.322875656

Solving
-1.5 + x = -1.322875656

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 = -1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.322875656 + 1.5
x = -1.322875656 + 1.5

Combine like terms: -1.322875656 + 1.5 = 0.177124344
x = 0.177124344

Simplifying
x = 0.177124344

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.822875656, 0.177124344}
Solution
x = {2.822875656, 0.177124344}