x^2+13x+31=0

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Solution for x^2+13x+31=0 equation:


Simplifying
x2 + 13x + 31 = 0

Reorder the terms:
31 + 13x + x2 = 0

Solving
31 + 13x + 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 + 13x + -31 + x2 = 0 + -31

Reorder the terms:
31 + -31 + 13x + x2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 13x + x2 = 0 + -31
13x + x2 = 0 + -31

Combine like terms: 0 + -31 = -31
13x + x2 = -31

The x term is 13x.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13x + 42.25 + x2 = -31 + 42.25

Reorder the terms:
42.25 + 13x + x2 = -31 + 42.25

Combine like terms: -31 + 42.25 = 11.25
42.25 + 13x + x2 = 11.25

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + 6.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

x + 6.5 = 3.354101966 Simplifying x + 6.5 = 3.354101966 Reorder the terms: 6.5 + x = 3.354101966 Solving 6.5 + x = 3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = 3.354101966 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 3.354101966 + -6.5 x = 3.354101966 + -6.5 Combine like terms: 3.354101966 + -6.5 = -3.145898034 x = -3.145898034 Simplifying x = -3.145898034

Subproblem 2

x + 6.5 = -3.354101966 Simplifying x + 6.5 = -3.354101966 Reorder the terms: 6.5 + x = -3.354101966 Solving 6.5 + x = -3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = -3.354101966 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -3.354101966 + -6.5 x = -3.354101966 + -6.5 Combine like terms: -3.354101966 + -6.5 = -9.854101966 x = -9.854101966 Simplifying x = -9.854101966

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.145898034, -9.854101966}

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