0=2x^2+16x+31

Solution for 0=2x^2+16x+31 equation:

Simplifying
0 = 2x2 + 16x + 31

Reorder the terms:
0 = 31 + 16x + 2x2

Solving
0 = 31 + 16x + 2x2

Solving for variable 'x'.

Combine like terms: 0 + -31 = -31
-31 + -16x + -2x2 = 31 + 16x + 2x2 + -31 + -16x + -2x2

Reorder the terms:
-31 + -16x + -2x2 = 31 + -31 + 16x + -16x + 2x2 + -2x2

Combine like terms: 31 + -31 = 0
-31 + -16x + -2x2 = 0 + 16x + -16x + 2x2 + -2x2
-31 + -16x + -2x2 = 16x + -16x + 2x2 + -2x2

Combine like terms: 16x + -16x = 0
-31 + -16x + -2x2 = 0 + 2x2 + -2x2
-31 + -16x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-31 + -16x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(31 + 16x + 2x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(31 + 16x + 2x2)' equal to zero and attempt to solve:

Simplifying
31 + 16x + 2x2 = 0

Solving
31 + 16x + 2x2 = 0

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

Divide each side by '2'.
15.5 + 8x + x2 = 0

Move the constant term to the right:

Add '-15.5' to each side of the equation.
15.5 + 8x + -15.5 + x2 = 0 + -15.5

Reorder the terms:
15.5 + -15.5 + 8x + x2 = 0 + -15.5

Combine like terms: 15.5 + -15.5 = 0.0
0.0 + 8x + x2 = 0 + -15.5
8x + x2 = 0 + -15.5

Combine like terms: 0 + -15.5 = -15.5
8x + x2 = -15.5

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -15.5 + 16

Reorder the terms:
16 + 8x + x2 = -15.5 + 16

Combine like terms: -15.5 + 16 = 0.5
16 + 8x + x2 = 0.5

Factor a perfect square on the left side:
(x + 4)(x + 4) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 4) equal to 0.707106781 and -0.707106781.

Subproblem 1
x + 4 = 0.707106781

Simplifying
x + 4 = 0.707106781

Reorder the terms:
4 + x = 0.707106781

Solving
4 + x = 0.707106781

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = 0.707106781 + -4

Combine like terms: 4 + -4 = 0
0 + x = 0.707106781 + -4
x = 0.707106781 + -4

Combine like terms: 0.707106781 + -4 = -3.292893219
x = -3.292893219

Simplifying
x = -3.292893219

Subproblem 2
x + 4 = -0.707106781

Simplifying
x + 4 = -0.707106781

Reorder the terms:
4 + x = -0.707106781

Solving
4 + x = -0.707106781

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = -0.707106781 + -4

Combine like terms: 4 + -4 = 0
0 + x = -0.707106781 + -4
x = -0.707106781 + -4

Combine like terms: -0.707106781 + -4 = -4.707106781
x = -4.707106781

Simplifying
x = -4.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.292893219, -4.707106781}
Solution
x = {-3.292893219, -4.707106781}