4x^2-4x-6=0

Solution for 4x^2-4x-6=0 equation:

Simplifying
4x2 + -4x + -6 = 0

Reorder the terms:
-6 + -4x + 4x2 = 0

Solving
-6 + -4x + 4x2 = 0

Solving for variable 'x'.

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

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + -2x + 2x2)' equal to zero and attempt to solve:

Simplifying
-3 + -2x + 2x2 = 0

Solving
-3 + -2x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + -1x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -1x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -1x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -1x + x2 = 0 + 1.5
-1x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-1x + x2 = 1.5

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 = 1.5 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + -1x + x2 = 1.75

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
x + -0.5 = 1.322875656

Simplifying
x + -0.5 = 1.322875656

Reorder the terms:
-0.5 + x = 1.322875656

Solving
-0.5 + x = 1.322875656

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 = 1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 1.322875656 + 0.5
x = 1.322875656 + 0.5

Combine like terms: 1.322875656 + 0.5 = 1.822875656
x = 1.822875656

Simplifying
x = 1.822875656

Subproblem 2
x + -0.5 = -1.322875656

Simplifying
x + -0.5 = -1.322875656

Reorder the terms:
-0.5 + x = -1.322875656

Solving
-0.5 + x = -1.322875656

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 = -1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -1.322875656 + 0.5
x = -1.322875656 + 0.5

Combine like terms: -1.322875656 + 0.5 = -0.822875656
x = -0.822875656

Simplifying
x = -0.822875656

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.822875656, -0.822875656}
Solution
x = {1.822875656, -0.822875656}