-3x^2=8x-12

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Solution for -3x^2=8x-12 equation:


Simplifying
-3x2 = 8x + -12

Reorder the terms:
-3x2 = -12 + 8x

Solving
-3x2 = -12 + 8x

Solving for variable 'x'.

Reorder the terms:
12 + -8x + -3x2 = -12 + 8x + 12 + -8x

Reorder the terms:
12 + -8x + -3x2 = -12 + 12 + 8x + -8x

Combine like terms: -12 + 12 = 0
12 + -8x + -3x2 = 0 + 8x + -8x
12 + -8x + -3x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
12 + -8x + -3x2 = 0

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

Divide each side by '-3'.
-4 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 2.666666667x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 2.666666667x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 2.666666667x + x2 = 0 + 4
2.666666667x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
2.666666667x + x2 = 4

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 = 4 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 4 + 1.777777780

Combine like terms: 4 + 1.777777780 = 5.77777778
1.777777780 + 2.666666667x + x2 = 5.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 5.77777778

Calculate the square root of the right side: 2.403700851

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 2.403700851 and -2.403700851.

Subproblem 1

x + 1.333333334 = 2.403700851 Simplifying x + 1.333333334 = 2.403700851 Reorder the terms: 1.333333334 + x = 2.403700851 Solving 1.333333334 + x = 2.403700851 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 = 2.403700851 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 2.403700851 + -1.333333334 x = 2.403700851 + -1.333333334 Combine like terms: 2.403700851 + -1.333333334 = 1.070367517 x = 1.070367517 Simplifying x = 1.070367517

Subproblem 2

x + 1.333333334 = -2.403700851 Simplifying x + 1.333333334 = -2.403700851 Reorder the terms: 1.333333334 + x = -2.403700851 Solving 1.333333334 + x = -2.403700851 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 = -2.403700851 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -2.403700851 + -1.333333334 x = -2.403700851 + -1.333333334 Combine like terms: -2.403700851 + -1.333333334 = -3.737034185 x = -3.737034185 Simplifying x = -3.737034185

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.070367517, -3.737034185}

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