x^2-27x-54=0

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Solution for x^2-27x-54=0 equation:


Simplifying
x2 + -27x + -54 = 0

Reorder the terms:
-54 + -27x + x2 = 0

Solving
-54 + -27x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -27x + 54 + x2 = 0 + 54

Reorder the terms:
-54 + 54 + -27x + x2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -27x + x2 = 0 + 54
-27x + x2 = 0 + 54

Combine like terms: 0 + 54 = 54
-27x + x2 = 54

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 = 54 + 182.25

Reorder the terms:
182.25 + -27x + x2 = 54 + 182.25

Combine like terms: 54 + 182.25 = 236.25
182.25 + -27x + x2 = 236.25

Factor a perfect square on the left side:
(x + -13.5)(x + -13.5) = 236.25

Calculate the square root of the right side: 15.370426149

Break this problem into two subproblems by setting 
(x + -13.5) equal to 15.370426149 and -15.370426149.

Subproblem 1

x + -13.5 = 15.370426149 Simplifying x + -13.5 = 15.370426149 Reorder the terms: -13.5 + x = 15.370426149 Solving -13.5 + x = 15.370426149 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 = 15.370426149 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x = 15.370426149 + 13.5 x = 15.370426149 + 13.5 Combine like terms: 15.370426149 + 13.5 = 28.870426149 x = 28.870426149 Simplifying x = 28.870426149

Subproblem 2

x + -13.5 = -15.370426149 Simplifying x + -13.5 = -15.370426149 Reorder the terms: -13.5 + x = -15.370426149 Solving -13.5 + x = -15.370426149 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 = -15.370426149 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + x = -15.370426149 + 13.5 x = -15.370426149 + 13.5 Combine like terms: -15.370426149 + 13.5 = -1.870426149 x = -1.870426149 Simplifying x = -1.870426149

Solution

The solution to the problem is based on the solutions from the subproblems. x = {28.870426149, -1.870426149}

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