(2x-1)(2x-1)=21

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Solution for (2x-1)(2x-1)=21 equation: 


Simplifying
(2x + -1)(2x + -1) = 21

Reorder the terms:
(-1 + 2x)(2x + -1) = 21

Reorder the terms:
(-1 + 2x)(-1 + 2x) = 21

Multiply (-1 + 2x) * (-1 + 2x)
(-1(-1 + 2x) + 2x * (-1 + 2x)) = 21
((-1 * -1 + 2x * -1) + 2x * (-1 + 2x)) = 21
((1 + -2x) + 2x * (-1 + 2x)) = 21
(1 + -2x + (-1 * 2x + 2x * 2x)) = 21
(1 + -2x + (-2x + 4x2)) = 21

Combine like terms: -2x + -2x = -4x
(1 + -4x + 4x2) = 21

Solving
1 + -4x + 4x2 = 21

Solving for variable 'x'.

Reorder the terms:
1 + -21 + -4x + 4x2 = 21 + -21

Combine like terms: 1 + -21 = -20
-20 + -4x + 4x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-20 + -4x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-5 + -1x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-5 + -1x + x2)' equal to zero and attempt to solve:

Simplifying
-5 + -1x + x2 = 0

Solving
-5 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + -1x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1x + x2 = 0 + 5
-1x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1x + x2 = 5

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1x + x2 = 5.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
x + -0.5 = 2.291287847

Simplifying
x + -0.5 = 2.291287847

Reorder the terms:
-0.5 + x = 2.291287847

Solving
-0.5 + x = 2.291287847

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 2.291287847 + 0.5
x = 2.291287847 + 0.5

Combine like terms: 2.291287847 + 0.5 = 2.791287847
x = 2.791287847

Simplifying
x = 2.791287847


Subproblem 2
x + -0.5 = -2.291287847

Simplifying
x + -0.5 = -2.291287847

Reorder the terms:
-0.5 + x = -2.291287847

Solving
-0.5 + x = -2.291287847

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -2.291287847 + 0.5
x = -2.291287847 + 0.5

Combine like terms: -2.291287847 + 0.5 = -1.791287847
x = -1.791287847

Simplifying
x = -1.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.791287847, -1.791287847}
Solution
x = {2.791287847, -1.791287847}

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