x^2=4x^2+8x-32

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Solution for x^2=4x^2+8x-32 equation:


Simplifying
x2 = 4x2 + 8x + -32

Reorder the terms:
x2 = -32 + 8x + 4x2

Solving
x2 = -32 + 8x + 4x2

Solving for variable 'x'.

Reorder the terms:
32 + -8x + x2 + -4x2 = -32 + 8x + 4x2 + 32 + -8x + -4x2

Combine like terms: x2 + -4x2 = -3x2
32 + -8x + -3x2 = -32 + 8x + 4x2 + 32 + -8x + -4x2

Reorder the terms:
32 + -8x + -3x2 = -32 + 32 + 8x + -8x + 4x2 + -4x2

Combine like terms: -32 + 32 = 0
32 + -8x + -3x2 = 0 + 8x + -8x + 4x2 + -4x2
32 + -8x + -3x2 = 8x + -8x + 4x2 + -4x2

Combine like terms: 8x + -8x = 0
32 + -8x + -3x2 = 0 + 4x2 + -4x2
32 + -8x + -3x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
32 + -8x + -3x2 = 0

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

Divide each side by '-3'.
-10.66666667 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '10.66666667' to each side of the equation.
-10.66666667 + 2.666666667x + 10.66666667 + x2 = 0 + 10.66666667

Reorder the terms:
-10.66666667 + 10.66666667 + 2.666666667x + x2 = 0 + 10.66666667

Combine like terms: -10.66666667 + 10.66666667 = 0.00000000
0.00000000 + 2.666666667x + x2 = 0 + 10.66666667
2.666666667x + x2 = 0 + 10.66666667

Combine like terms: 0 + 10.66666667 = 10.66666667
2.666666667x + x2 = 10.66666667

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = 10.66666667 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 10.66666667 + 1.777777780

Combine like terms: 10.66666667 + 1.777777780 = 12.44444445
1.777777780 + 2.666666667x + x2 = 12.44444445

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 12.44444445

Calculate the square root of the right side: 3.527668416

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 3.527668416 and -3.527668416.

Subproblem 1

x + 1.333333334 = 3.527668416 Simplifying x + 1.333333334 = 3.527668416 Reorder the terms: 1.333333334 + x = 3.527668416 Solving 1.333333334 + x = 3.527668416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 3.527668416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 3.527668416 + -1.333333334 x = 3.527668416 + -1.333333334 Combine like terms: 3.527668416 + -1.333333334 = 2.194335082 x = 2.194335082 Simplifying x = 2.194335082

Subproblem 2

x + 1.333333334 = -3.527668416 Simplifying x + 1.333333334 = -3.527668416 Reorder the terms: 1.333333334 + x = -3.527668416 Solving 1.333333334 + x = -3.527668416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -3.527668416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -3.527668416 + -1.333333334 x = -3.527668416 + -1.333333334 Combine like terms: -3.527668416 + -1.333333334 = -4.86100175 x = -4.86100175 Simplifying x = -4.86100175

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.194335082, -4.86100175}

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