31x^2+12x-5=0

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Solution for 31x^2+12x-5=0 equation: 


Simplifying
31x2 + 12x + -5 = 0

Reorder the terms:
-5 + 12x + 31x2 = 0

Solving
-5 + 12x + 31x2 = 0

Solving for variable 'x'.

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

Divide each side by '31'.
-0.1612903226 + 0.3870967742x + x2 = 0

Move the constant term to the right:

Add '0.1612903226' to each side of the equation.
-0.1612903226 + 0.3870967742x + 0.1612903226 + x2 = 0 + 0.1612903226

Reorder the terms:
-0.1612903226 + 0.1612903226 + 0.3870967742x + x2 = 0 + 0.1612903226

Combine like terms: -0.1612903226 + 0.1612903226 = 0.0000000000
0.0000000000 + 0.3870967742x + x2 = 0 + 0.1612903226
0.3870967742x + x2 = 0 + 0.1612903226

Combine like terms: 0 + 0.1612903226 = 0.1612903226
0.3870967742x + x2 = 0.1612903226

The x term is 0.3870967742x.  Take half its coefficient (0.1935483871).
Square it (0.03746097815) and add it to both sides.

Add '0.03746097815' to each side of the equation.
0.3870967742x + 0.03746097815 + x2 = 0.1612903226 + 0.03746097815

Reorder the terms:
0.03746097815 + 0.3870967742x + x2 = 0.1612903226 + 0.03746097815

Combine like terms: 0.1612903226 + 0.03746097815 = 0.19875130075
0.03746097815 + 0.3870967742x + x2 = 0.19875130075

Factor a perfect square on the left side:
(x + 0.1935483871)(x + 0.1935483871) = 0.19875130075

Calculate the square root of the right side: 0.445815321

Break this problem into two subproblems by setting 
(x + 0.1935483871) equal to 0.445815321 and -0.445815321.


Subproblem 1
x + 0.1935483871 = 0.445815321

Simplifying
x + 0.1935483871 = 0.445815321

Reorder the terms:
0.1935483871 + x = 0.445815321

Solving
0.1935483871 + x = 0.445815321

Solving for variable 'x'.

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

Add '-0.1935483871' to each side of the equation.
0.1935483871 + -0.1935483871 + x = 0.445815321 + -0.1935483871

Combine like terms: 0.1935483871 + -0.1935483871 = 0.0000000000
0.0000000000 + x = 0.445815321 + -0.1935483871
x = 0.445815321 + -0.1935483871

Combine like terms: 0.445815321 + -0.1935483871 = 0.2522669339
x = 0.2522669339

Simplifying
x = 0.2522669339


Subproblem 2
x + 0.1935483871 = -0.445815321

Simplifying
x + 0.1935483871 = -0.445815321

Reorder the terms:
0.1935483871 + x = -0.445815321

Solving
0.1935483871 + x = -0.445815321

Solving for variable 'x'.

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

Add '-0.1935483871' to each side of the equation.
0.1935483871 + -0.1935483871 + x = -0.445815321 + -0.1935483871

Combine like terms: 0.1935483871 + -0.1935483871 = 0.0000000000
0.0000000000 + x = -0.445815321 + -0.1935483871
x = -0.445815321 + -0.1935483871

Combine like terms: -0.445815321 + -0.1935483871 = -0.6393637081
x = -0.6393637081

Simplifying
x = -0.6393637081


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.2522669339, -0.6393637081}

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