2x^2+31x-8.5=0

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Solution for 2x^2+31x-8.5=0 equation: 


Simplifying
2x2 + 31x + -8.5 = 0

Reorder the terms:
-8.5 + 31x + 2x2 = 0

Solving
-8.5 + 31x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-4.25 + 15.5x + x2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + 15.5x + 4.25 + x2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + 15.5x + x2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + 15.5x + x2 = 0 + 4.25
15.5x + x2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
15.5x + x2 = 4.25

The x term is 15.5x.  Take half its coefficient (7.75).
Square it (60.0625) and add it to both sides.

Add '60.0625' to each side of the equation.
15.5x + 60.0625 + x2 = 4.25 + 60.0625

Reorder the terms:
60.0625 + 15.5x + x2 = 4.25 + 60.0625

Combine like terms: 4.25 + 60.0625 = 64.3125
60.0625 + 15.5x + x2 = 64.3125

Factor a perfect square on the left side:
(x + 7.75)(x + 7.75) = 64.3125

Calculate the square root of the right side: 8.019507466

Break this problem into two subproblems by setting 
(x + 7.75) equal to 8.019507466 and -8.019507466.


Subproblem 1
x + 7.75 = 8.019507466

Simplifying
x + 7.75 = 8.019507466

Reorder the terms:
7.75 + x = 8.019507466

Solving
7.75 + x = 8.019507466

Solving for variable 'x'.

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

Add '-7.75' to each side of the equation.
7.75 + -7.75 + x = 8.019507466 + -7.75

Combine like terms: 7.75 + -7.75 = 0.00
0.00 + x = 8.019507466 + -7.75
x = 8.019507466 + -7.75

Combine like terms: 8.019507466 + -7.75 = 0.269507466
x = 0.269507466

Simplifying
x = 0.269507466


Subproblem 2
x + 7.75 = -8.019507466

Simplifying
x + 7.75 = -8.019507466

Reorder the terms:
7.75 + x = -8.019507466

Solving
7.75 + x = -8.019507466

Solving for variable 'x'.

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

Add '-7.75' to each side of the equation.
7.75 + -7.75 + x = -8.019507466 + -7.75

Combine like terms: 7.75 + -7.75 = 0.00
0.00 + x = -8.019507466 + -7.75
x = -8.019507466 + -7.75

Combine like terms: -8.019507466 + -7.75 = -15.769507466
x = -15.769507466

Simplifying
x = -15.769507466


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.269507466, -15.769507466}

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