(2x-5)(2x-5)-11=0

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Solution for (2x-5)(2x-5)-11=0 equation: 


Simplifying
(2x + -5)(2x + -5) + -11 = 0

Reorder the terms:
(-5 + 2x)(2x + -5) + -11 = 0

Reorder the terms:
(-5 + 2x)(-5 + 2x) + -11 = 0

Multiply (-5 + 2x) * (-5 + 2x)
(-5(-5 + 2x) + 2x * (-5 + 2x)) + -11 = 0
((-5 * -5 + 2x * -5) + 2x * (-5 + 2x)) + -11 = 0
((25 + -10x) + 2x * (-5 + 2x)) + -11 = 0
(25 + -10x + (-5 * 2x + 2x * 2x)) + -11 = 0
(25 + -10x + (-10x + 4x2)) + -11 = 0

Combine like terms: -10x + -10x = -20x
(25 + -20x + 4x2) + -11 = 0

Reorder the terms:
25 + -11 + -20x + 4x2 = 0

Combine like terms: 25 + -11 = 14
14 + -20x + 4x2 = 0

Solving
14 + -20x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(7 + -10x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(7 + -10x + 2x2)' equal to zero and attempt to solve:

Simplifying
7 + -10x + 2x2 = 0

Solving
7 + -10x + 2x2 = 0

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

Divide each side by '2'.
3.5 + -5x + x2 = 0

Move the constant term to the right:

Add '-3.5' to each side of the equation.
3.5 + -5x + -3.5 + x2 = 0 + -3.5

Reorder the terms:
3.5 + -3.5 + -5x + x2 = 0 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + -5x + x2 = 0 + -3.5
-5x + x2 = 0 + -3.5

Combine like terms: 0 + -3.5 = -3.5
-5x + x2 = -3.5

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -3.5 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -3.5 + 6.25

Combine like terms: -3.5 + 6.25 = 2.75
6.25 + -5x + x2 = 2.75

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(x + -2.5) equal to 1.658312395 and -1.658312395.


Subproblem 1
x + -2.5 = 1.658312395

Simplifying
x + -2.5 = 1.658312395

Reorder the terms:
-2.5 + x = 1.658312395

Solving
-2.5 + x = 1.658312395

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 1.658312395 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 1.658312395 + 2.5
x = 1.658312395 + 2.5

Combine like terms: 1.658312395 + 2.5 = 4.158312395
x = 4.158312395

Simplifying
x = 4.158312395


Subproblem 2
x + -2.5 = -1.658312395

Simplifying
x + -2.5 = -1.658312395

Reorder the terms:
-2.5 + x = -1.658312395

Solving
-2.5 + x = -1.658312395

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -1.658312395 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -1.658312395 + 2.5
x = -1.658312395 + 2.5

Combine like terms: -1.658312395 + 2.5 = 0.841687605
x = 0.841687605

Simplifying
x = 0.841687605


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.158312395, 0.841687605}
Solution
x = {4.158312395, 0.841687605}

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