(4x-2)(3x+4)=(2x-3)(2x-3)

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Solution for (4x-2)(3x+4)=(2x-3)(2x-3) equation:


Simplifying
(4x + -2)(3x + 4) = (2x + -3)(2x + -3)

Reorder the terms:
(-2 + 4x)(3x + 4) = (2x + -3)(2x + -3)

Reorder the terms:
(-2 + 4x)(4 + 3x) = (2x + -3)(2x + -3)

Multiply (-2 + 4x) * (4 + 3x)
(-2(4 + 3x) + 4x * (4 + 3x)) = (2x + -3)(2x + -3)
((4 * -2 + 3x * -2) + 4x * (4 + 3x)) = (2x + -3)(2x + -3)
((-8 + -6x) + 4x * (4 + 3x)) = (2x + -3)(2x + -3)
(-8 + -6x + (4 * 4x + 3x * 4x)) = (2x + -3)(2x + -3)
(-8 + -6x + (16x + 12x2)) = (2x + -3)(2x + -3)

Combine like terms: -6x + 16x = 10x
(-8 + 10x + 12x2) = (2x + -3)(2x + -3)

Reorder the terms:
-8 + 10x + 12x2 = (-3 + 2x)(2x + -3)

Reorder the terms:
-8 + 10x + 12x2 = (-3 + 2x)(-3 + 2x)

Multiply (-3 + 2x) * (-3 + 2x)
-8 + 10x + 12x2 = (-3(-3 + 2x) + 2x * (-3 + 2x))
-8 + 10x + 12x2 = ((-3 * -3 + 2x * -3) + 2x * (-3 + 2x))
-8 + 10x + 12x2 = ((9 + -6x) + 2x * (-3 + 2x))
-8 + 10x + 12x2 = (9 + -6x + (-3 * 2x + 2x * 2x))
-8 + 10x + 12x2 = (9 + -6x + (-6x + 4x2))

Combine like terms: -6x + -6x = -12x
-8 + 10x + 12x2 = (9 + -12x + 4x2)

Solving
-8 + 10x + 12x2 = 9 + -12x + 4x2

Solving for variable 'x'.

Reorder the terms:
-8 + -9 + 10x + 12x + 12x2 + -4x2 = 9 + -12x + 4x2 + -9 + 12x + -4x2

Combine like terms: -8 + -9 = -17
-17 + 10x + 12x + 12x2 + -4x2 = 9 + -12x + 4x2 + -9 + 12x + -4x2

Combine like terms: 10x + 12x = 22x
-17 + 22x + 12x2 + -4x2 = 9 + -12x + 4x2 + -9 + 12x + -4x2

Combine like terms: 12x2 + -4x2 = 8x2
-17 + 22x + 8x2 = 9 + -12x + 4x2 + -9 + 12x + -4x2

Reorder the terms:
-17 + 22x + 8x2 = 9 + -9 + -12x + 12x + 4x2 + -4x2

Combine like terms: 9 + -9 = 0
-17 + 22x + 8x2 = 0 + -12x + 12x + 4x2 + -4x2
-17 + 22x + 8x2 = -12x + 12x + 4x2 + -4x2

Combine like terms: -12x + 12x = 0
-17 + 22x + 8x2 = 0 + 4x2 + -4x2
-17 + 22x + 8x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-17 + 22x + 8x2 = 0

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

Divide each side by '8'.
-2.125 + 2.75x + x2 = 0

Move the constant term to the right:

Add '2.125' to each side of the equation.
-2.125 + 2.75x + 2.125 + x2 = 0 + 2.125

Reorder the terms:
-2.125 + 2.125 + 2.75x + x2 = 0 + 2.125

Combine like terms: -2.125 + 2.125 = 0.000
0.000 + 2.75x + x2 = 0 + 2.125
2.75x + x2 = 0 + 2.125

Combine like terms: 0 + 2.125 = 2.125
2.75x + x2 = 2.125

The x term is 2.75x.  Take half its coefficient (1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
2.75x + 1.890625 + x2 = 2.125 + 1.890625

Reorder the terms:
1.890625 + 2.75x + x2 = 2.125 + 1.890625

Combine like terms: 2.125 + 1.890625 = 4.015625
1.890625 + 2.75x + x2 = 4.015625

Factor a perfect square on the left side:
(x + 1.375)(x + 1.375) = 4.015625

Calculate the square root of the right side: 2.003902443

Break this problem into two subproblems by setting 
(x + 1.375) equal to 2.003902443 and -2.003902443.

Subproblem 1

x + 1.375 = 2.003902443 Simplifying x + 1.375 = 2.003902443 Reorder the terms: 1.375 + x = 2.003902443 Solving 1.375 + x = 2.003902443 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 2.003902443 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 2.003902443 + -1.375 x = 2.003902443 + -1.375 Combine like terms: 2.003902443 + -1.375 = 0.628902443 x = 0.628902443 Simplifying x = 0.628902443

Subproblem 2

x + 1.375 = -2.003902443 Simplifying x + 1.375 = -2.003902443 Reorder the terms: 1.375 + x = -2.003902443 Solving 1.375 + x = -2.003902443 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -2.003902443 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -2.003902443 + -1.375 x = -2.003902443 + -1.375 Combine like terms: -2.003902443 + -1.375 = -3.378902443 x = -3.378902443 Simplifying x = -3.378902443

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.628902443, -3.378902443}

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