12x^2-90x+120=0

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Solution for 12x^2-90x+120=0 equation: 


Simplifying
12x2 + -90x + 120 = 0

Reorder the terms:
120 + -90x + 12x2 = 0

Solving
120 + -90x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(20 + -15x + 2x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(20 + -15x + 2x2)' equal to zero and attempt to solve:

Simplifying
20 + -15x + 2x2 = 0

Solving
20 + -15x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
10 + -7.5x + x2 = 0

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + -7.5x + -10 + x2 = 0 + -10

Reorder the terms:
10 + -10 + -7.5x + x2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -7.5x + x2 = 0 + -10
-7.5x + x2 = 0 + -10

Combine like terms: 0 + -10 = -10
-7.5x + x2 = -10

The x term is -7.5x.  Take half its coefficient (-3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
-7.5x + 14.0625 + x2 = -10 + 14.0625

Reorder the terms:
14.0625 + -7.5x + x2 = -10 + 14.0625

Combine like terms: -10 + 14.0625 = 4.0625
14.0625 + -7.5x + x2 = 4.0625

Factor a perfect square on the left side:
(x + -3.75)(x + -3.75) = 4.0625

Calculate the square root of the right side: 2.015564437

Break this problem into two subproblems by setting 
(x + -3.75) equal to 2.015564437 and -2.015564437.


Subproblem 1
x + -3.75 = 2.015564437

Simplifying
x + -3.75 = 2.015564437

Reorder the terms:
-3.75 + x = 2.015564437

Solving
-3.75 + x = 2.015564437

Solving for variable 'x'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + x = 2.015564437 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + x = 2.015564437 + 3.75
x = 2.015564437 + 3.75

Combine like terms: 2.015564437 + 3.75 = 5.765564437
x = 5.765564437

Simplifying
x = 5.765564437


Subproblem 2
x + -3.75 = -2.015564437

Simplifying
x + -3.75 = -2.015564437

Reorder the terms:
-3.75 + x = -2.015564437

Solving
-3.75 + x = -2.015564437

Solving for variable 'x'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + x = -2.015564437 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + x = -2.015564437 + 3.75
x = -2.015564437 + 3.75

Combine like terms: -2.015564437 + 3.75 = 1.734435563
x = 1.734435563

Simplifying
x = 1.734435563


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.765564437, 1.734435563}
Solution
x = {5.765564437, 1.734435563}

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