x^2+x-108=0

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Solution for x^2+x-108=0 equation:


Simplifying
x2 + x + -108 = 0

Reorder the terms:
-108 + x + x2 = 0

Solving
-108 + x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + x + 108 + x2 = 0 + 108

Reorder the terms:
-108 + 108 + x + x2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + x + x2 = 0 + 108
x + x2 = 0 + 108

Combine like terms: 0 + 108 = 108
x + x2 = 108

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 108 + 0.25

Reorder the terms:
0.25 + x + x2 = 108 + 0.25

Combine like terms: 108 + 0.25 = 108.25
0.25 + x + x2 = 108.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 108.25

Calculate the square root of the right side: 10.404326023

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.404326023 and -10.404326023.

Subproblem 1

x + 0.5 = 10.404326023 Simplifying x + 0.5 = 10.404326023 Reorder the terms: 0.5 + x = 10.404326023 Solving 0.5 + x = 10.404326023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 10.404326023 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 10.404326023 + -0.5 x = 10.404326023 + -0.5 Combine like terms: 10.404326023 + -0.5 = 9.904326023 x = 9.904326023 Simplifying x = 9.904326023

Subproblem 2

x + 0.5 = -10.404326023 Simplifying x + 0.5 = -10.404326023 Reorder the terms: 0.5 + x = -10.404326023 Solving 0.5 + x = -10.404326023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -10.404326023 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -10.404326023 + -0.5 x = -10.404326023 + -0.5 Combine like terms: -10.404326023 + -0.5 = -10.904326023 x = -10.904326023 Simplifying x = -10.904326023

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.904326023, -10.904326023}

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