(2x-1)(2x-4)=x(x+2)

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Solution for (2x-1)(2x-4)=x(x+2) equation:


Simplifying
(2x + -1)(2x + -4) = x(x + 2)

Reorder the terms:
(-1 + 2x)(2x + -4) = x(x + 2)

Reorder the terms:
(-1 + 2x)(-4 + 2x) = x(x + 2)

Multiply (-1 + 2x) * (-4 + 2x)
(-1(-4 + 2x) + 2x * (-4 + 2x)) = x(x + 2)
((-4 * -1 + 2x * -1) + 2x * (-4 + 2x)) = x(x + 2)
((4 + -2x) + 2x * (-4 + 2x)) = x(x + 2)
(4 + -2x + (-4 * 2x + 2x * 2x)) = x(x + 2)
(4 + -2x + (-8x + 4x2)) = x(x + 2)

Combine like terms: -2x + -8x = -10x
(4 + -10x + 4x2) = x(x + 2)

Reorder the terms:
4 + -10x + 4x2 = x(2 + x)
4 + -10x + 4x2 = (2 * x + x * x)
4 + -10x + 4x2 = (2x + x2)

Solving
4 + -10x + 4x2 = 2x + x2

Solving for variable 'x'.

Reorder the terms:
4 + -10x + -2x + 4x2 + -1x2 = 2x + x2 + -2x + -1x2

Combine like terms: -10x + -2x = -12x
4 + -12x + 4x2 + -1x2 = 2x + x2 + -2x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
4 + -12x + 3x2 = 2x + x2 + -2x + -1x2

Reorder the terms:
4 + -12x + 3x2 = 2x + -2x + x2 + -1x2

Combine like terms: 2x + -2x = 0
4 + -12x + 3x2 = 0 + x2 + -1x2
4 + -12x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
4 + -12x + 3x2 = 0

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

Divide each side by '3'.
1.333333333 + -4x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -4x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -4x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -4x + x2 = 0 + -1.333333333
-4x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-4x + x2 = -1.333333333

The x term is -4x.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4x + 4 + x2 = -1.333333333 + 4

Reorder the terms:
4 + -4x + x2 = -1.333333333 + 4

Combine like terms: -1.333333333 + 4 = 2.666666667
4 + -4x + x2 = 2.666666667

Factor a perfect square on the left side:
(x + -2)(x + -2) = 2.666666667

Calculate the square root of the right side: 1.632993162

Break this problem into two subproblems by setting 
(x + -2) equal to 1.632993162 and -1.632993162.

Subproblem 1

x + -2 = 1.632993162 Simplifying x + -2 = 1.632993162 Reorder the terms: -2 + x = 1.632993162 Solving -2 + x = 1.632993162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + x = 1.632993162 + 2 x = 1.632993162 + 2 Combine like terms: 1.632993162 + 2 = 3.632993162 x = 3.632993162 Simplifying x = 3.632993162

Subproblem 2

x + -2 = -1.632993162 Simplifying x + -2 = -1.632993162 Reorder the terms: -2 + x = -1.632993162 Solving -2 + x = -1.632993162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + x = -1.632993162 + 2 x = -1.632993162 + 2 Combine like terms: -1.632993162 + 2 = 0.367006838 x = 0.367006838 Simplifying x = 0.367006838

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.632993162, 0.367006838}

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