12x^2-84x+80=0

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Solution for 12x^2-84x+80=0 equation: 


Simplifying
12x2 + -84x + 80 = 0

Reorder the terms:
80 + -84x + 12x2 = 0

Solving
80 + -84x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(20 + -21x + 3x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(20 + -21x + 3x2)' equal to zero and attempt to solve:

Simplifying
20 + -21x + 3x2 = 0

Solving
20 + -21x + 3x2 = 0

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

Divide each side by '3'.
6.666666667 + -7x + x2 = 0

Move the constant term to the right:

Add '-6.666666667' to each side of the equation.
6.666666667 + -7x + -6.666666667 + x2 = 0 + -6.666666667

Reorder the terms:
6.666666667 + -6.666666667 + -7x + x2 = 0 + -6.666666667

Combine like terms: 6.666666667 + -6.666666667 = 0.000000000
0.000000000 + -7x + x2 = 0 + -6.666666667
-7x + x2 = 0 + -6.666666667

Combine like terms: 0 + -6.666666667 = -6.666666667
-7x + x2 = -6.666666667

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -6.666666667 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -6.666666667 + 12.25

Combine like terms: -6.666666667 + 12.25 = 5.583333333
12.25 + -7x + x2 = 5.583333333

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 5.583333333

Calculate the square root of the right side: 2.362907813

Break this problem into two subproblems by setting 
(x + -3.5) equal to 2.362907813 and -2.362907813.


Subproblem 1
x + -3.5 = 2.362907813

Simplifying
x + -3.5 = 2.362907813

Reorder the terms:
-3.5 + x = 2.362907813

Solving
-3.5 + x = 2.362907813

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 2.362907813 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 2.362907813 + 3.5
x = 2.362907813 + 3.5

Combine like terms: 2.362907813 + 3.5 = 5.862907813
x = 5.862907813

Simplifying
x = 5.862907813


Subproblem 2
x + -3.5 = -2.362907813

Simplifying
x + -3.5 = -2.362907813

Reorder the terms:
-3.5 + x = -2.362907813

Solving
-3.5 + x = -2.362907813

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -2.362907813 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -2.362907813 + 3.5
x = -2.362907813 + 3.5

Combine like terms: -2.362907813 + 3.5 = 1.137092187
x = 1.137092187

Simplifying
x = 1.137092187


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.862907813, 1.137092187}
Solution
x = {5.862907813, 1.137092187}

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