15x^2+31x+12=0

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Solution for 15x^2+31x+12=0 equation:


Simplifying
15x2 + 31x + 12 = 0

Reorder the terms:
12 + 31x + 15x2 = 0

Solving
12 + 31x + 15x2 = 0

Solving for variable 'x'.

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

Divide each side by '15'.
0.8 + 2.066666667x + x2 = 0

Move the constant term to the right:

Add '-0.8' to each side of the equation.
0.8 + 2.066666667x + -0.8 + x2 = 0 + -0.8

Reorder the terms:
0.8 + -0.8 + 2.066666667x + x2 = 0 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + 2.066666667x + x2 = 0 + -0.8
2.066666667x + x2 = 0 + -0.8

Combine like terms: 0 + -0.8 = -0.8
2.066666667x + x2 = -0.8

The x term is 2.066666667x.  Take half its coefficient (1.033333334).
Square it (1.067777779) and add it to both sides.

Add '1.067777779' to each side of the equation.
2.066666667x + 1.067777779 + x2 = -0.8 + 1.067777779

Reorder the terms:
1.067777779 + 2.066666667x + x2 = -0.8 + 1.067777779

Combine like terms: -0.8 + 1.067777779 = 0.267777779
1.067777779 + 2.066666667x + x2 = 0.267777779

Factor a perfect square on the left side:
(x + 1.033333334)(x + 1.033333334) = 0.267777779

Calculate the square root of the right side: 0.517472491

Break this problem into two subproblems by setting 
(x + 1.033333334) equal to 0.517472491 and -0.517472491.

Subproblem 1

x + 1.033333334 = 0.517472491 Simplifying x + 1.033333334 = 0.517472491 Reorder the terms: 1.033333334 + x = 0.517472491 Solving 1.033333334 + x = 0.517472491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = 0.517472491 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = 0.517472491 + -1.033333334 x = 0.517472491 + -1.033333334 Combine like terms: 0.517472491 + -1.033333334 = -0.515860843 x = -0.515860843 Simplifying x = -0.515860843

Subproblem 2

x + 1.033333334 = -0.517472491 Simplifying x + 1.033333334 = -0.517472491 Reorder the terms: 1.033333334 + x = -0.517472491 Solving 1.033333334 + x = -0.517472491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = -0.517472491 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = -0.517472491 + -1.033333334 x = -0.517472491 + -1.033333334 Combine like terms: -0.517472491 + -1.033333334 = -1.550805825 x = -1.550805825 Simplifying x = -1.550805825

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.515860843, -1.550805825}

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