6x^2-5x-6=x

Solution for 6x^2-5x-6=x equation:

Simplifying
6x2 + -5x + -6 = x

Reorder the terms:
-6 + -5x + 6x2 = x

Solving
-6 + -5x + 6x2 = x

Solving for variable 'x'.

Reorder the terms:
-6 + -5x + -1x + 6x2 = x + -1x

Combine like terms: -5x + -1x = -6x
-6 + -6x + 6x2 = x + -1x

Combine like terms: x + -1x = 0
-6 + -6x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-1 + -1x + x2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(-1 + -1x + x2)' equal to zero and attempt to solve:

Simplifying
-1 + -1x + x2 = 0

Solving
-1 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 1 + 0.25

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

Combine like terms: 1 + 0.25 = 1.25
0.25 + -1x + x2 = 1.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x + -0.5 = 1.118033989

Simplifying
x + -0.5 = 1.118033989

Reorder the terms:
-0.5 + x = 1.118033989

Solving
-0.5 + x = 1.118033989

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 = 1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 1.118033989 + 0.5
x = 1.118033989 + 0.5

Combine like terms: 1.118033989 + 0.5 = 1.618033989
x = 1.618033989

Simplifying
x = 1.618033989

Subproblem 2
x + -0.5 = -1.118033989

Simplifying
x + -0.5 = -1.118033989

Reorder the terms:
-0.5 + x = -1.118033989

Solving
-0.5 + x = -1.118033989

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 = -1.118033989 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -1.118033989 + 0.5
x = -1.118033989 + 0.5

Combine like terms: -1.118033989 + 0.5 = -0.618033989
x = -0.618033989

Simplifying
x = -0.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.618033989, -0.618033989}
Solution
x = {1.618033989, -0.618033989}