12x^2-108x+180=0

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Solution for 12x^2-108x+180=0 equation: 


Simplifying
12x2 + -108x + 180 = 0

Reorder the terms:
180 + -108x + 12x2 = 0

Solving
180 + -108x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '12'.
12(15 + -9x + x2) = 0

Ignore the factor 12.

Subproblem 1
Set the factor '(15 + -9x + x2)' equal to zero and attempt to solve:

Simplifying
15 + -9x + x2 = 0

Solving
15 + -9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -9x + -15 + x2 = 0 + -15

Reorder the terms:
15 + -15 + -9x + x2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -9x + x2 = 0 + -15
-9x + x2 = 0 + -15

Combine like terms: 0 + -15 = -15
-9x + x2 = -15

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = -15 + 20.25

Reorder the terms:
20.25 + -9x + x2 = -15 + 20.25

Combine like terms: -15 + 20.25 = 5.25
20.25 + -9x + x2 = 5.25

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -4.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
x + -4.5 = 2.291287847

Simplifying
x + -4.5 = 2.291287847

Reorder the terms:
-4.5 + x = 2.291287847

Solving
-4.5 + x = 2.291287847

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = 2.291287847 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = 2.291287847 + 4.5
x = 2.291287847 + 4.5

Combine like terms: 2.291287847 + 4.5 = 6.791287847
x = 6.791287847

Simplifying
x = 6.791287847


Subproblem 2
x + -4.5 = -2.291287847

Simplifying
x + -4.5 = -2.291287847

Reorder the terms:
-4.5 + x = -2.291287847

Solving
-4.5 + x = -2.291287847

Solving for variable 'x'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + x = -2.291287847 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + x = -2.291287847 + 4.5
x = -2.291287847 + 4.5

Combine like terms: -2.291287847 + 4.5 = 2.208712153
x = 2.208712153

Simplifying
x = 2.208712153


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.791287847, 2.208712153}
Solution
x = {6.791287847, 2.208712153}

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