12x^2-80x+84=0

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


Simplifying
12x2 + -80x + 84 = 0

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

Solving
84 + -80x + 12x2 = 0

Solving for variable 'x'.

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

Ignore the factor 4.

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

Simplifying
21 + -20x + 3x2 = 0

Solving
21 + -20x + 3x2 = 0

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

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

Move the constant term to the right:

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

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

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

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

The x term is -6.666666667x.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667x + 11.11111112 + x2 = -7 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667x + x2 = -7 + 11.11111112

Combine like terms: -7 + 11.11111112 = 4.11111112
11.11111112 + -6.666666667x + x2 = 4.11111112

Factor a perfect square on the left side:
(x + -3.333333334)(x + -3.333333334) = 4.11111112

Calculate the square root of the right side: 2.027587512

Break this problem into two subproblems by setting 
(x + -3.333333334) equal to 2.027587512 and -2.027587512.


Subproblem 1
x + -3.333333334 = 2.027587512

Simplifying
x + -3.333333334 = 2.027587512

Reorder the terms:
-3.333333334 + x = 2.027587512

Solving
-3.333333334 + x = 2.027587512

Solving for variable 'x'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = 2.027587512 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = 2.027587512 + 3.333333334
x = 2.027587512 + 3.333333334

Combine like terms: 2.027587512 + 3.333333334 = 5.360920846
x = 5.360920846

Simplifying
x = 5.360920846


Subproblem 2
x + -3.333333334 = -2.027587512

Simplifying
x + -3.333333334 = -2.027587512

Reorder the terms:
-3.333333334 + x = -2.027587512

Solving
-3.333333334 + x = -2.027587512

Solving for variable 'x'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = -2.027587512 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = -2.027587512 + 3.333333334
x = -2.027587512 + 3.333333334

Combine like terms: -2.027587512 + 3.333333334 = 1.305745822
x = 1.305745822

Simplifying
x = 1.305745822


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.360920846, 1.305745822}
Solution
x = {5.360920846, 1.305745822}

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