q^2+14q+31=0

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Solution for q^2+14q+31=0 equation:


Simplifying
q2 + 14q + 31 = 0

Reorder the terms:
31 + 14q + q2 = 0

Solving
31 + 14q + q2 = 0

Solving for variable 'q'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 14q + -31 + q2 = 0 + -31

Reorder the terms:
31 + -31 + 14q + q2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 14q + q2 = 0 + -31
14q + q2 = 0 + -31

Combine like terms: 0 + -31 = -31
14q + q2 = -31

The q term is 14q.  Take half its coefficient (7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
14q + 49 + q2 = -31 + 49

Reorder the terms:
49 + 14q + q2 = -31 + 49

Combine like terms: -31 + 49 = 18
49 + 14q + q2 = 18

Factor a perfect square on the left side:
(q + 7)(q + 7) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(q + 7) equal to 4.242640687 and -4.242640687.

Subproblem 1

q + 7 = 4.242640687 Simplifying q + 7 = 4.242640687 Reorder the terms: 7 + q = 4.242640687 Solving 7 + q = 4.242640687 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + q = 4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + q = 4.242640687 + -7 q = 4.242640687 + -7 Combine like terms: 4.242640687 + -7 = -2.757359313 q = -2.757359313 Simplifying q = -2.757359313

Subproblem 2

q + 7 = -4.242640687 Simplifying q + 7 = -4.242640687 Reorder the terms: 7 + q = -4.242640687 Solving 7 + q = -4.242640687 Solving for variable 'q'. Move all terms containing q to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + q = -4.242640687 + -7 Combine like terms: 7 + -7 = 0 0 + q = -4.242640687 + -7 q = -4.242640687 + -7 Combine like terms: -4.242640687 + -7 = -11.242640687 q = -11.242640687 Simplifying q = -11.242640687

Solution

The solution to the problem is based on the solutions from the subproblems. q = {-2.757359313, -11.242640687}

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