(2x+4)(2x-3)+5x=10x

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Solution for (2x+4)(2x-3)+5x=10x equation:


Simplifying
(2x + 4)(2x + -3) + 5x = 10x

Reorder the terms:
(4 + 2x)(2x + -3) + 5x = 10x

Reorder the terms:
(4 + 2x)(-3 + 2x) + 5x = 10x

Multiply (4 + 2x) * (-3 + 2x)
(4(-3 + 2x) + 2x * (-3 + 2x)) + 5x = 10x
((-3 * 4 + 2x * 4) + 2x * (-3 + 2x)) + 5x = 10x
((-12 + 8x) + 2x * (-3 + 2x)) + 5x = 10x
(-12 + 8x + (-3 * 2x + 2x * 2x)) + 5x = 10x
(-12 + 8x + (-6x + 4x2)) + 5x = 10x

Combine like terms: 8x + -6x = 2x
(-12 + 2x + 4x2) + 5x = 10x

Reorder the terms:
-12 + 2x + 5x + 4x2 = 10x

Combine like terms: 2x + 5x = 7x
-12 + 7x + 4x2 = 10x

Solving
-12 + 7x + 4x2 = 10x

Solving for variable 'x'.

Reorder the terms:
-12 + 7x + -10x + 4x2 = 10x + -10x

Combine like terms: 7x + -10x = -3x
-12 + -3x + 4x2 = 10x + -10x

Combine like terms: 10x + -10x = 0
-12 + -3x + 4x2 = 0

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

Divide each side by '4'.
-3 + -0.75x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.75x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.75x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.75x + x2 = 0 + 3
-0.75x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.75x + x2 = 3

The x term is -0.75x.  Take half its coefficient (-0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
-0.75x + 0.140625 + x2 = 3 + 0.140625

Reorder the terms:
0.140625 + -0.75x + x2 = 3 + 0.140625

Combine like terms: 3 + 0.140625 = 3.140625
0.140625 + -0.75x + x2 = 3.140625

Factor a perfect square on the left side:
(x + -0.375)(x + -0.375) = 3.140625

Calculate the square root of the right side: 1.77218086

Break this problem into two subproblems by setting 
(x + -0.375) equal to 1.77218086 and -1.77218086.

Subproblem 1

x + -0.375 = 1.77218086 Simplifying x + -0.375 = 1.77218086 Reorder the terms: -0.375 + x = 1.77218086 Solving -0.375 + x = 1.77218086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 1.77218086 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.77218086 + 0.375 x = 1.77218086 + 0.375 Combine like terms: 1.77218086 + 0.375 = 2.14718086 x = 2.14718086 Simplifying x = 2.14718086

Subproblem 2

x + -0.375 = -1.77218086 Simplifying x + -0.375 = -1.77218086 Reorder the terms: -0.375 + x = -1.77218086 Solving -0.375 + x = -1.77218086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -1.77218086 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.77218086 + 0.375 x = -1.77218086 + 0.375 Combine like terms: -1.77218086 + 0.375 = -1.39718086 x = -1.39718086 Simplifying x = -1.39718086

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.14718086, -1.39718086}

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