x^2-27x-180=0

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Solution for x^2-27x-180=0 equation:


Simplifying
x2 + -27x + -180 = 0

Reorder the terms:
-180 + -27x + x2 = 0

Solving
-180 + -27x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + -27x + 180 + x2 = 0 + 180

Reorder the terms:
-180 + 180 + -27x + x2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + -27x + x2 = 0 + 180
-27x + x2 = 0 + 180

Combine like terms: 0 + 180 = 180
-27x + x2 = 180

The x term is -27x.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27x + 182.25 + x2 = 180 + 182.25

Reorder the terms:
182.25 + -27x + x2 = 180 + 182.25

Combine like terms: 180 + 182.25 = 362.25
182.25 + -27x + x2 = 362.25

Factor a perfect square on the left side:
(x + -13.5)(x + -13.5) = 362.25

Calculate the square root of the right side: 19.032866311

Break this problem into two subproblems by setting 
(x + -13.5) equal to 19.032866311 and -19.032866311.

Subproblem 1

x + -13.5 = 19.032866311 Simplifying x + -13.5 = 19.032866311 Reorder the terms: -13.5 + x = 19.032866311 Solving -13.5 + x = 19.032866311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x = 19.032866311 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x = 19.032866311 + 13.5 x = 19.032866311 + 13.5 Combine like terms: 19.032866311 + 13.5 = 32.532866311 x = 32.532866311 Simplifying x = 32.532866311

Subproblem 2

x + -13.5 = -19.032866311 Simplifying x + -13.5 = -19.032866311 Reorder the terms: -13.5 + x = -19.032866311 Solving -13.5 + x = -19.032866311 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.5' to each side of the equation. -13.5 + 13.5 + x = -19.032866311 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x = -19.032866311 + 13.5 x = -19.032866311 + 13.5 Combine like terms: -19.032866311 + 13.5 = -5.532866311 x = -5.532866311 Simplifying x = -5.532866311

Solution

The solution to the problem is based on the solutions from the subproblems. x = {32.532866311, -5.532866311}

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