6x^2-20x-126=0

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Solution for 6x^2-20x-126=0 equation: 


Simplifying
6x2 + -20x + -126 = 0

Reorder the terms:
-126 + -20x + 6x2 = 0

Solving
-126 + -20x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-63 + -10x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-63 + -10x + 3x2)' equal to zero and attempt to solve:

Simplifying
-63 + -10x + 3x2 = 0

Solving
-63 + -10x + 3x2 = 0

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

Divide each side by '3'.
-21 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -3.333333333x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + -3.333333333x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -3.333333333x + x2 = 0 + 21
-3.333333333x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
-3.333333333x + x2 = 21

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = 21 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = 21 + 2.777777779

Combine like terms: 21 + 2.777777779 = 23.777777779
2.777777779 + -3.333333333x + x2 = 23.777777779

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 23.777777779

Calculate the square root of the right side: 4.87624628

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 4.87624628 and -4.87624628.


Subproblem 1
x + -1.666666667 = 4.87624628

Simplifying
x + -1.666666667 = 4.87624628

Reorder the terms:
-1.666666667 + x = 4.87624628

Solving
-1.666666667 + x = 4.87624628

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = 4.87624628 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = 4.87624628 + 1.666666667
x = 4.87624628 + 1.666666667

Combine like terms: 4.87624628 + 1.666666667 = 6.542912947
x = 6.542912947

Simplifying
x = 6.542912947


Subproblem 2
x + -1.666666667 = -4.87624628

Simplifying
x + -1.666666667 = -4.87624628

Reorder the terms:
-1.666666667 + x = -4.87624628

Solving
-1.666666667 + x = -4.87624628

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = -4.87624628 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = -4.87624628 + 1.666666667
x = -4.87624628 + 1.666666667

Combine like terms: -4.87624628 + 1.666666667 = -3.209579613
x = -3.209579613

Simplifying
x = -3.209579613


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.542912947, -3.209579613}
Solution
x = {6.542912947, -3.209579613}

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