(2x-3)(2x-3)-8=0

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Solution for (2x-3)(2x-3)-8=0 equation:


Simplifying
(2x + -3)(2x + -3) + -8 = 0

Reorder the terms:
(-3 + 2x)(2x + -3) + -8 = 0

Reorder the terms:
(-3 + 2x)(-3 + 2x) + -8 = 0

Multiply (-3 + 2x) * (-3 + 2x)
(-3(-3 + 2x) + 2x * (-3 + 2x)) + -8 = 0
((-3 * -3 + 2x * -3) + 2x * (-3 + 2x)) + -8 = 0
((9 + -6x) + 2x * (-3 + 2x)) + -8 = 0
(9 + -6x + (-3 * 2x + 2x * 2x)) + -8 = 0
(9 + -6x + (-6x + 4x2)) + -8 = 0

Combine like terms: -6x + -6x = -12x
(9 + -12x + 4x2) + -8 = 0

Reorder the terms:
9 + -8 + -12x + 4x2 = 0

Combine like terms: 9 + -8 = 1
1 + -12x + 4x2 = 0

Solving
1 + -12x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
0.25 + -3x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -3x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -3x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -3x + x2 = 0 + -0.25
-3x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-3x + x2 = -0.25

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 = -0.25 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -0.25 + 2.25

Combine like terms: -0.25 + 2.25 = 2
2.25 + -3x + x2 = 2

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -1.5 = 1.414213562 Simplifying x + -1.5 = 1.414213562 Reorder the terms: -1.5 + x = 1.414213562 Solving -1.5 + x = 1.414213562 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 = 1.414213562 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.414213562 + 1.5 x = 1.414213562 + 1.5 Combine like terms: 1.414213562 + 1.5 = 2.914213562 x = 2.914213562 Simplifying x = 2.914213562

Subproblem 2

x + -1.5 = -1.414213562 Simplifying x + -1.5 = -1.414213562 Reorder the terms: -1.5 + x = -1.414213562 Solving -1.5 + x = -1.414213562 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 = -1.414213562 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.414213562 + 1.5 x = -1.414213562 + 1.5 Combine like terms: -1.414213562 + 1.5 = 0.085786438 x = 0.085786438 Simplifying x = 0.085786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.914213562, 0.085786438}

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