(4x^2-2x)+(5+2x^2-2x-8)=0

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Solution for (4x^2-2x)+(5+2x^2-2x-8)=0 equation:


Simplifying
(4x2 + -2x) + (5 + 2x2 + -2x + -8) = 0

Reorder the terms:
(-2x + 4x2) + (5 + 2x2 + -2x + -8) = 0

Remove parenthesis around (-2x + 4x2)
-2x + 4x2 + (5 + 2x2 + -2x + -8) = 0

Reorder the terms:
-2x + 4x2 + (5 + -8 + -2x + 2x2) = 0

Combine like terms: 5 + -8 = -3
-2x + 4x2 + (-3 + -2x + 2x2) = 0

Remove parenthesis around (-3 + -2x + 2x2)
-2x + 4x2 + -3 + -2x + 2x2 = 0

Reorder the terms:
-3 + -2x + -2x + 4x2 + 2x2 = 0

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

Combine like terms: 4x2 + 2x2 = 6x2
-3 + -4x + 6x2 = 0

Solving
-3 + -4x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-0.5 + -0.6666666667x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -0.6666666667x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -0.6666666667x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -0.6666666667x + x2 = 0 + 0.5
-0.6666666667x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-0.6666666667x + x2 = 0.5

The x term is -0.6666666667x.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667x + 0.1111111112 + x2 = 0.5 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667x + x2 = 0.5 + 0.1111111112

Combine like terms: 0.5 + 0.1111111112 = 0.6111111112
0.1111111112 + -0.6666666667x + x2 = 0.6111111112

Factor a perfect square on the left side:
(x + -0.3333333334)(x + -0.3333333334) = 0.6111111112

Calculate the square root of the right side: 0.78173596

Break this problem into two subproblems by setting 
(x + -0.3333333334) equal to 0.78173596 and -0.78173596.

Subproblem 1

x + -0.3333333334 = 0.78173596 Simplifying x + -0.3333333334 = 0.78173596 Reorder the terms: -0.3333333334 + x = 0.78173596 Solving -0.3333333334 + x = 0.78173596 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 0.78173596 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 0.78173596 + 0.3333333334 x = 0.78173596 + 0.3333333334 Combine like terms: 0.78173596 + 0.3333333334 = 1.1150692934 x = 1.1150692934 Simplifying x = 1.1150692934

Subproblem 2

x + -0.3333333334 = -0.78173596 Simplifying x + -0.3333333334 = -0.78173596 Reorder the terms: -0.3333333334 + x = -0.78173596 Solving -0.3333333334 + x = -0.78173596 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -0.78173596 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -0.78173596 + 0.3333333334 x = -0.78173596 + 0.3333333334 Combine like terms: -0.78173596 + 0.3333333334 = -0.4484026266 x = -0.4484026266 Simplifying x = -0.4484026266

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.1150692934, -0.4484026266}

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