6x^2-78x-27=0

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Solution for 6x^2-78x-27=0 equation: 


Simplifying
6x2 + -78x + -27 = 0

Reorder the terms:
-27 + -78x + 6x2 = 0

Solving
-27 + -78x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-9 + -26x + 2x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-9 + -26x + 2x2)' equal to zero and attempt to solve:

Simplifying
-9 + -26x + 2x2 = 0

Solving
-9 + -26x + 2x2 = 0

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

Divide each side by '2'.
-4.5 + -13x + x2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + -13x + 4.5 + x2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + -13x + x2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + -13x + x2 = 0 + 4.5
-13x + x2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
-13x + x2 = 4.5

The x term is -13x.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x + 42.25 + x2 = 4.5 + 42.25

Reorder the terms:
42.25 + -13x + x2 = 4.5 + 42.25

Combine like terms: 4.5 + 42.25 = 46.75
42.25 + -13x + x2 = 46.75

Factor a perfect square on the left side:
(x + -6.5)(x + -6.5) = 46.75

Calculate the square root of the right side: 6.837397166

Break this problem into two subproblems by setting 
(x + -6.5) equal to 6.837397166 and -6.837397166.


Subproblem 1
x + -6.5 = 6.837397166

Simplifying
x + -6.5 = 6.837397166

Reorder the terms:
-6.5 + x = 6.837397166

Solving
-6.5 + x = 6.837397166

Solving for variable 'x'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + x = 6.837397166 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + x = 6.837397166 + 6.5
x = 6.837397166 + 6.5

Combine like terms: 6.837397166 + 6.5 = 13.337397166
x = 13.337397166

Simplifying
x = 13.337397166


Subproblem 2
x + -6.5 = -6.837397166

Simplifying
x + -6.5 = -6.837397166

Reorder the terms:
-6.5 + x = -6.837397166

Solving
-6.5 + x = -6.837397166

Solving for variable 'x'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + x = -6.837397166 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + x = -6.837397166 + 6.5
x = -6.837397166 + 6.5

Combine like terms: -6.837397166 + 6.5 = -0.337397166
x = -0.337397166

Simplifying
x = -0.337397166


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {13.337397166, -0.337397166}
Solution
x = {13.337397166, -0.337397166}

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