6x^2-6x-22=0

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Solution for 6x^2-6x-22=0 equation: 


Simplifying
6x2 + -6x + -22 = 0

Reorder the terms:
-22 + -6x + 6x2 = 0

Solving
-22 + -6x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-11 + -3x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-11 + -3x + 3x2)' equal to zero and attempt to solve:

Simplifying
-11 + -3x + 3x2 = 0

Solving
-11 + -3x + 3x2 = 0

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

Divide each side by '3'.
-3.666666667 + -1x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -1x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -1x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -1x + x2 = 0 + 3.666666667
-1x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-1x + x2 = 3.666666667

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 3.666666667 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 3.666666667 + 0.25

Combine like terms: 3.666666667 + 0.25 = 3.916666667
0.25 + -1x + x2 = 3.916666667

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 3.916666667

Calculate the square root of the right side: 1.979057015

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.979057015 and -1.979057015.


Subproblem 1
x + -0.5 = 1.979057015

Simplifying
x + -0.5 = 1.979057015

Reorder the terms:
-0.5 + x = 1.979057015

Solving
-0.5 + x = 1.979057015

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 1.979057015 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 1.979057015 + 0.5
x = 1.979057015 + 0.5

Combine like terms: 1.979057015 + 0.5 = 2.479057015
x = 2.479057015

Simplifying
x = 2.479057015


Subproblem 2
x + -0.5 = -1.979057015

Simplifying
x + -0.5 = -1.979057015

Reorder the terms:
-0.5 + x = -1.979057015

Solving
-0.5 + x = -1.979057015

Solving for variable 'x'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -1.979057015 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -1.979057015 + 0.5
x = -1.979057015 + 0.5

Combine like terms: -1.979057015 + 0.5 = -1.479057015
x = -1.479057015

Simplifying
x = -1.479057015


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.479057015, -1.479057015}
Solution
x = {2.479057015, -1.479057015}

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