g^2+12g+31=0

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Solution for g^2+12g+31=0 equation:


Simplifying
g2 + 12g + 31 = 0

Reorder the terms:
31 + 12g + g2 = 0

Solving
31 + 12g + g2 = 0

Solving for variable 'g'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 12g + -31 + g2 = 0 + -31

Reorder the terms:
31 + -31 + 12g + g2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 12g + g2 = 0 + -31
12g + g2 = 0 + -31

Combine like terms: 0 + -31 = -31
12g + g2 = -31

The g term is 12g.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12g + 36 + g2 = -31 + 36

Reorder the terms:
36 + 12g + g2 = -31 + 36

Combine like terms: -31 + 36 = 5
36 + 12g + g2 = 5

Factor a perfect square on the left side:
(g + 6)(g + 6) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(g + 6) equal to 2.236067978 and -2.236067978.

Subproblem 1

g + 6 = 2.236067978 Simplifying g + 6 = 2.236067978 Reorder the terms: 6 + g = 2.236067978 Solving 6 + g = 2.236067978 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + g = 2.236067978 + -6 Combine like terms: 6 + -6 = 0 0 + g = 2.236067978 + -6 g = 2.236067978 + -6 Combine like terms: 2.236067978 + -6 = -3.763932022 g = -3.763932022 Simplifying g = -3.763932022

Subproblem 2

g + 6 = -2.236067978 Simplifying g + 6 = -2.236067978 Reorder the terms: 6 + g = -2.236067978 Solving 6 + g = -2.236067978 Solving for variable 'g'. Move all terms containing g to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + g = -2.236067978 + -6 Combine like terms: 6 + -6 = 0 0 + g = -2.236067978 + -6 g = -2.236067978 + -6 Combine like terms: -2.236067978 + -6 = -8.236067978 g = -8.236067978 Simplifying g = -8.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. g = {-3.763932022, -8.236067978}

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