x^2-21x-180=0

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Solution for x^2-21x-180=0 equation: 


Simplifying
x2 + -21x + -180 = 0

Reorder the terms:
-180 + -21x + x2 = 0

Solving
-180 + -21x + 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 + -21x + 180 + x2 = 0 + 180

Reorder the terms:
-180 + 180 + -21x + x2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + -21x + x2 = 0 + 180
-21x + x2 = 0 + 180

Combine like terms: 0 + 180 = 180
-21x + x2 = 180

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = 180 + 110.25

Reorder the terms:
110.25 + -21x + x2 = 180 + 110.25

Combine like terms: 180 + 110.25 = 290.25
110.25 + -21x + x2 = 290.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 290.25

Calculate the square root of the right side: 17.036725037

Break this problem into two subproblems by setting 
(x + -10.5) equal to 17.036725037 and -17.036725037.


Subproblem 1
x + -10.5 = 17.036725037

Simplifying
x + -10.5 = 17.036725037

Reorder the terms:
-10.5 + x = 17.036725037

Solving
-10.5 + x = 17.036725037

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = 17.036725037 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = 17.036725037 + 10.5
x = 17.036725037 + 10.5

Combine like terms: 17.036725037 + 10.5 = 27.536725037
x = 27.536725037

Simplifying
x = 27.536725037


Subproblem 2
x + -10.5 = -17.036725037

Simplifying
x + -10.5 = -17.036725037

Reorder the terms:
-10.5 + x = -17.036725037

Solving
-10.5 + x = -17.036725037

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '10.5' to each side of the equation.
-10.5 + 10.5 + x = -17.036725037 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + x = -17.036725037 + 10.5
x = -17.036725037 + 10.5

Combine like terms: -17.036725037 + 10.5 = -6.536725037
x = -6.536725037

Simplifying
x = -6.536725037


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {27.536725037, -6.536725037}

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