2x^2-6+2x+x^2+12=78

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Solution for 2x^2-6+2x+x^2+12=78 equation:


Simplifying
2x2 + -6 + 2x + x2 + 12 = 78

Reorder the terms:
-6 + 12 + 2x + 2x2 + x2 = 78

Combine like terms: -6 + 12 = 6
6 + 2x + 2x2 + x2 = 78

Combine like terms: 2x2 + x2 = 3x2
6 + 2x + 3x2 = 78

Solving
6 + 2x + 3x2 = 78

Solving for variable 'x'.

Reorder the terms:
6 + -78 + 2x + 3x2 = 78 + -78

Combine like terms: 6 + -78 = -72
-72 + 2x + 3x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-72 + 2x + 3x2 = 0

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

Divide each side by '3'.
-24 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + 0.6666666667x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + 0.6666666667x + x2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + 0.6666666667x + x2 = 0 + 24
0.6666666667x + x2 = 0 + 24

Combine like terms: 0 + 24 = 24
0.6666666667x + x2 = 24

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 = 24 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 24 + 0.1111111112

Combine like terms: 24 + 0.1111111112 = 24.1111111112
0.1111111112 + 0.6666666667x + x2 = 24.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 24.1111111112

Calculate the square root of the right side: 4.910306621

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 4.910306621 and -4.910306621.

Subproblem 1

x + 0.3333333334 = 4.910306621 Simplifying x + 0.3333333334 = 4.910306621 Reorder the terms: 0.3333333334 + x = 4.910306621 Solving 0.3333333334 + x = 4.910306621 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 = 4.910306621 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 4.910306621 + -0.3333333334 x = 4.910306621 + -0.3333333334 Combine like terms: 4.910306621 + -0.3333333334 = 4.5769732876 x = 4.5769732876 Simplifying x = 4.5769732876

Subproblem 2

x + 0.3333333334 = -4.910306621 Simplifying x + 0.3333333334 = -4.910306621 Reorder the terms: 0.3333333334 + x = -4.910306621 Solving 0.3333333334 + x = -4.910306621 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 = -4.910306621 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -4.910306621 + -0.3333333334 x = -4.910306621 + -0.3333333334 Combine like terms: -4.910306621 + -0.3333333334 = -5.2436399544 x = -5.2436399544 Simplifying x = -5.2436399544

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.5769732876, -5.2436399544}

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