12x^2-76x+84=0

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Solution for 12x^2-76x+84=0 equation: 


Simplifying
12x2 + -76x + 84 = 0

Reorder the terms:
84 + -76x + 12x2 = 0

Solving
84 + -76x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(21 + -19x + 3x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(21 + -19x + 3x2)' equal to zero and attempt to solve:

Simplifying
21 + -19x + 3x2 = 0

Solving
21 + -19x + 3x2 = 0

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

Divide each side by '3'.
7 + -6.333333333x + x2 = 0

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + -6.333333333x + -7 + x2 = 0 + -7

Reorder the terms:
7 + -7 + -6.333333333x + x2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + -6.333333333x + x2 = 0 + -7
-6.333333333x + x2 = 0 + -7

Combine like terms: 0 + -7 = -7
-6.333333333x + x2 = -7

The x term is -6.333333333x.  Take half its coefficient (-3.166666667).
Square it (10.02777778) and add it to both sides.

Add '10.02777778' to each side of the equation.
-6.333333333x + 10.02777778 + x2 = -7 + 10.02777778

Reorder the terms:
10.02777778 + -6.333333333x + x2 = -7 + 10.02777778

Combine like terms: -7 + 10.02777778 = 3.02777778
10.02777778 + -6.333333333x + x2 = 3.02777778

Factor a perfect square on the left side:
(x + -3.166666667)(x + -3.166666667) = 3.02777778

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(x + -3.166666667) equal to 1.740051085 and -1.740051085.


Subproblem 1
x + -3.166666667 = 1.740051085

Simplifying
x + -3.166666667 = 1.740051085

Reorder the terms:
-3.166666667 + x = 1.740051085

Solving
-3.166666667 + x = 1.740051085

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = 1.740051085 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = 1.740051085 + 3.166666667
x = 1.740051085 + 3.166666667

Combine like terms: 1.740051085 + 3.166666667 = 4.906717752
x = 4.906717752

Simplifying
x = 4.906717752


Subproblem 2
x + -3.166666667 = -1.740051085

Simplifying
x + -3.166666667 = -1.740051085

Reorder the terms:
-3.166666667 + x = -1.740051085

Solving
-3.166666667 + x = -1.740051085

Solving for variable 'x'.

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

Add '3.166666667' to each side of the equation.
-3.166666667 + 3.166666667 + x = -1.740051085 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + x = -1.740051085 + 3.166666667
x = -1.740051085 + 3.166666667

Combine like terms: -1.740051085 + 3.166666667 = 1.426615582
x = 1.426615582

Simplifying
x = 1.426615582


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.906717752, 1.426615582}
Solution
x = {4.906717752, 1.426615582}

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