x^2+18x=-31

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Solution for x^2+18x=-31 equation:


Simplifying
x2 + 18x = -31

Reorder the terms:
18x + x2 = -31

Solving
18x + x2 = -31

Solving for variable 'x'.

Reorder the terms:
31 + 18x + x2 = -31 + 31

Combine like terms: -31 + 31 = 0
31 + 18x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 18x + -31 + x2 = 0 + -31

Reorder the terms:
31 + -31 + 18x + x2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 18x + x2 = 0 + -31
18x + x2 = 0 + -31

Combine like terms: 0 + -31 = -31
18x + x2 = -31

The x term is 18x.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18x + 81 + x2 = -31 + 81

Reorder the terms:
81 + 18x + x2 = -31 + 81

Combine like terms: -31 + 81 = 50
81 + 18x + x2 = 50

Factor a perfect square on the left side:
(x + 9)(x + 9) = 50

Calculate the square root of the right side: 7.071067812

Break this problem into two subproblems by setting 
(x + 9) equal to 7.071067812 and -7.071067812.

Subproblem 1

x + 9 = 7.071067812 Simplifying x + 9 = 7.071067812 Reorder the terms: 9 + x = 7.071067812 Solving 9 + x = 7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + x = 7.071067812 + -9 Combine like terms: 9 + -9 = 0 0 + x = 7.071067812 + -9 x = 7.071067812 + -9 Combine like terms: 7.071067812 + -9 = -1.928932188 x = -1.928932188 Simplifying x = -1.928932188

Subproblem 2

x + 9 = -7.071067812 Simplifying x + 9 = -7.071067812 Reorder the terms: 9 + x = -7.071067812 Solving 9 + x = -7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + x = -7.071067812 + -9 Combine like terms: 9 + -9 = 0 0 + x = -7.071067812 + -9 x = -7.071067812 + -9 Combine like terms: -7.071067812 + -9 = -16.071067812 x = -16.071067812 Simplifying x = -16.071067812

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.928932188, -16.071067812}

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