x^2-3x-108=180

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Solution for x^2-3x-108=180 equation:


Simplifying
x2 + -3x + -108 = 180

Reorder the terms:
-108 + -3x + x2 = 180

Solving
-108 + -3x + x2 = 180

Solving for variable 'x'.

Reorder the terms:
-108 + -180 + -3x + x2 = 180 + -180

Combine like terms: -108 + -180 = -288
-288 + -3x + x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-288 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '288' to each side of the equation.
-288 + -3x + 288 + x2 = 0 + 288

Reorder the terms:
-288 + 288 + -3x + x2 = 0 + 288

Combine like terms: -288 + 288 = 0
0 + -3x + x2 = 0 + 288
-3x + x2 = 0 + 288

Combine like terms: 0 + 288 = 288
-3x + x2 = 288

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 288 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 288 + 2.25

Combine like terms: 288 + 2.25 = 290.25
2.25 + -3x + x2 = 290.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 290.25

Calculate the square root of the right side: 17.036725037

Break this problem into two subproblems by setting 
(x + -1.5) equal to 17.036725037 and -17.036725037.

Subproblem 1

x + -1.5 = 17.036725037 Simplifying x + -1.5 = 17.036725037 Reorder the terms: -1.5 + x = 17.036725037 Solving -1.5 + x = 17.036725037 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 17.036725037 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 17.036725037 + 1.5 x = 17.036725037 + 1.5 Combine like terms: 17.036725037 + 1.5 = 18.536725037 x = 18.536725037 Simplifying x = 18.536725037

Subproblem 2

x + -1.5 = -17.036725037 Simplifying x + -1.5 = -17.036725037 Reorder the terms: -1.5 + x = -17.036725037 Solving -1.5 + x = -17.036725037 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -17.036725037 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -17.036725037 + 1.5 x = -17.036725037 + 1.5 Combine like terms: -17.036725037 + 1.5 = -15.536725037 x = -15.536725037 Simplifying x = -15.536725037

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.536725037, -15.536725037}

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