6x^2+31x+15=0

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Solution for 6x^2+31x+15=0 equation: 


Simplifying
6x2 + 31x + 15 = 0

Reorder the terms:
15 + 31x + 6x2 = 0

Solving
15 + 31x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
2.5 + 5.166666667x + x2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 5.166666667x + -2.5 + x2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 5.166666667x + x2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 5.166666667x + x2 = 0 + -2.5
5.166666667x + x2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
5.166666667x + x2 = -2.5

The x term is 5.166666667x.  Take half its coefficient (2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
5.166666667x + 6.673611115 + x2 = -2.5 + 6.673611115

Reorder the terms:
6.673611115 + 5.166666667x + x2 = -2.5 + 6.673611115

Combine like terms: -2.5 + 6.673611115 = 4.173611115
6.673611115 + 5.166666667x + x2 = 4.173611115

Factor a perfect square on the left side:
(x + 2.583333334)(x + 2.583333334) = 4.173611115

Calculate the square root of the right side: 2.04294178

Break this problem into two subproblems by setting 
(x + 2.583333334) equal to 2.04294178 and -2.04294178.


Subproblem 1
x + 2.583333334 = 2.04294178

Simplifying
x + 2.583333334 = 2.04294178

Reorder the terms:
2.583333334 + x = 2.04294178

Solving
2.583333334 + x = 2.04294178

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.583333334' to each side of the equation.
2.583333334 + -2.583333334 + x = 2.04294178 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + x = 2.04294178 + -2.583333334
x = 2.04294178 + -2.583333334

Combine like terms: 2.04294178 + -2.583333334 = -0.540391554
x = -0.540391554

Simplifying
x = -0.540391554


Subproblem 2
x + 2.583333334 = -2.04294178

Simplifying
x + 2.583333334 = -2.04294178

Reorder the terms:
2.583333334 + x = -2.04294178

Solving
2.583333334 + x = -2.04294178

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-2.583333334' to each side of the equation.
2.583333334 + -2.583333334 + x = -2.04294178 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + x = -2.04294178 + -2.583333334
x = -2.04294178 + -2.583333334

Combine like terms: -2.04294178 + -2.583333334 = -4.626275114
x = -4.626275114

Simplifying
x = -4.626275114


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.540391554, -4.626275114}

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