3x^2=2(5x-3x^2+12)

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Solution for 3x^2=2(5x-3x^2+12) equation:


Simplifying
3x2 = 2(5x + -3x2 + 12)

Reorder the terms:
3x2 = 2(12 + 5x + -3x2)
3x2 = (12 * 2 + 5x * 2 + -3x2 * 2)
3x2 = (24 + 10x + -6x2)

Solving
3x2 = 24 + 10x + -6x2

Solving for variable 'x'.

Reorder the terms:
-24 + -10x + 3x2 + 6x2 = 24 + 10x + -6x2 + -24 + -10x + 6x2

Combine like terms: 3x2 + 6x2 = 9x2
-24 + -10x + 9x2 = 24 + 10x + -6x2 + -24 + -10x + 6x2

Reorder the terms:
-24 + -10x + 9x2 = 24 + -24 + 10x + -10x + -6x2 + 6x2

Combine like terms: 24 + -24 = 0
-24 + -10x + 9x2 = 0 + 10x + -10x + -6x2 + 6x2
-24 + -10x + 9x2 = 10x + -10x + -6x2 + 6x2

Combine like terms: 10x + -10x = 0
-24 + -10x + 9x2 = 0 + -6x2 + 6x2
-24 + -10x + 9x2 = -6x2 + 6x2

Combine like terms: -6x2 + 6x2 = 0
-24 + -10x + 9x2 = 0

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

Divide each side by '9'.
-2.666666667 + -1.111111111x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -1.111111111x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -1.111111111x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -1.111111111x + x2 = 0 + 2.666666667
-1.111111111x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-1.111111111x + x2 = 2.666666667

The x term is -1.111111111x.  Take half its coefficient (-0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
-1.111111111x + 0.3086419752 + x2 = 2.666666667 + 0.3086419752

Reorder the terms:
0.3086419752 + -1.111111111x + x2 = 2.666666667 + 0.3086419752

Combine like terms: 2.666666667 + 0.3086419752 = 2.9753086422
0.3086419752 + -1.111111111x + x2 = 2.9753086422

Factor a perfect square on the left side:
(x + -0.5555555555)(x + -0.5555555555) = 2.9753086422

Calculate the square root of the right side: 1.7249083

Break this problem into two subproblems by setting 
(x + -0.5555555555) equal to 1.7249083 and -1.7249083.

Subproblem 1

x + -0.5555555555 = 1.7249083 Simplifying x + -0.5555555555 = 1.7249083 Reorder the terms: -0.5555555555 + x = 1.7249083 Solving -0.5555555555 + x = 1.7249083 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + x = 1.7249083 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + x = 1.7249083 + 0.5555555555 x = 1.7249083 + 0.5555555555 Combine like terms: 1.7249083 + 0.5555555555 = 2.2804638555 x = 2.2804638555 Simplifying x = 2.2804638555

Subproblem 2

x + -0.5555555555 = -1.7249083 Simplifying x + -0.5555555555 = -1.7249083 Reorder the terms: -0.5555555555 + x = -1.7249083 Solving -0.5555555555 + x = -1.7249083 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + x = -1.7249083 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + x = -1.7249083 + 0.5555555555 x = -1.7249083 + 0.5555555555 Combine like terms: -1.7249083 + 0.5555555555 = -1.1693527445 x = -1.1693527445 Simplifying x = -1.1693527445

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.2804638555, -1.1693527445}

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