x^2+27x+31=0

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Solution for x^2+27x+31=0 equation: 


Simplifying
x2 + 27x + 31 = 0

Reorder the terms:
31 + 27x + x2 = 0

Solving
31 + 27x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 27x + -31 + x2 = 0 + -31

Reorder the terms:
31 + -31 + 27x + x2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 27x + x2 = 0 + -31
27x + x2 = 0 + -31

Combine like terms: 0 + -31 = -31
27x + x2 = -31

The x term is 27x.  Take half its coefficient (13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
27x + 182.25 + x2 = -31 + 182.25

Reorder the terms:
182.25 + 27x + x2 = -31 + 182.25

Combine like terms: -31 + 182.25 = 151.25
182.25 + 27x + x2 = 151.25

Factor a perfect square on the left side:
(x + 13.5)(x + 13.5) = 151.25

Calculate the square root of the right side: 12.298373876

Break this problem into two subproblems by setting 
(x + 13.5) equal to 12.298373876 and -12.298373876.


Subproblem 1
x + 13.5 = 12.298373876

Simplifying
x + 13.5 = 12.298373876

Reorder the terms:
13.5 + x = 12.298373876

Solving
13.5 + x = 12.298373876

Solving for variable 'x'.

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

Add '-13.5' to each side of the equation.
13.5 + -13.5 + x = 12.298373876 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + x = 12.298373876 + -13.5
x = 12.298373876 + -13.5

Combine like terms: 12.298373876 + -13.5 = -1.201626124
x = -1.201626124

Simplifying
x = -1.201626124


Subproblem 2
x + 13.5 = -12.298373876

Simplifying
x + 13.5 = -12.298373876

Reorder the terms:
13.5 + x = -12.298373876

Solving
13.5 + x = -12.298373876

Solving for variable 'x'.

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

Add '-13.5' to each side of the equation.
13.5 + -13.5 + x = -12.298373876 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + x = -12.298373876 + -13.5
x = -12.298373876 + -13.5

Combine like terms: -12.298373876 + -13.5 = -25.798373876
x = -25.798373876

Simplifying
x = -25.798373876


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.201626124, -25.798373876}

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