x^2+2x=108

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


Simplifying
x2 + 2x = 108

Reorder the terms:
2x + x2 = 108

Solving
2x + x2 = 108

Solving for variable 'x'.

Reorder the terms:
-108 + 2x + x2 = 108 + -108

Combine like terms: 108 + -108 = 0
-108 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + 2x + 108 + x2 = 0 + 108

Reorder the terms:
-108 + 108 + 2x + x2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + 2x + x2 = 0 + 108
2x + x2 = 0 + 108

Combine like terms: 0 + 108 = 108
2x + x2 = 108

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 108 + 1

Reorder the terms:
1 + 2x + x2 = 108 + 1

Combine like terms: 108 + 1 = 109
1 + 2x + x2 = 109

Factor a perfect square on the left side:
(x + 1)(x + 1) = 109

Calculate the square root of the right side: 10.440306509

Break this problem into two subproblems by setting 
(x + 1) equal to 10.440306509 and -10.440306509.

Subproblem 1

x + 1 = 10.440306509 Simplifying x + 1 = 10.440306509 Reorder the terms: 1 + x = 10.440306509 Solving 1 + x = 10.440306509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 10.440306509 + -1 Combine like terms: 1 + -1 = 0 0 + x = 10.440306509 + -1 x = 10.440306509 + -1 Combine like terms: 10.440306509 + -1 = 9.440306509 x = 9.440306509 Simplifying x = 9.440306509

Subproblem 2

x + 1 = -10.440306509 Simplifying x + 1 = -10.440306509 Reorder the terms: 1 + x = -10.440306509 Solving 1 + x = -10.440306509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -10.440306509 + -1 Combine like terms: 1 + -1 = 0 0 + x = -10.440306509 + -1 x = -10.440306509 + -1 Combine like terms: -10.440306509 + -1 = -11.440306509 x = -11.440306509 Simplifying x = -11.440306509

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.440306509, -11.440306509}

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