0=4x^2-x-6

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

Simplifying
0 = 4x2 + -1x + -6

Reorder the terms:
0 = -6 + -1x + 4x2

Solving
0 = -6 + -1x + 4x2

Solving for variable 'x'.

Combine like terms: 0 + 6 = 6
6 + x + -4x2 = -6 + -1x + 4x2 + 6 + x + -4x2

Reorder the terms:
6 + x + -4x2 = -6 + 6 + -1x + x + 4x2 + -4x2

Combine like terms: -6 + 6 = 0
6 + x + -4x2 = 0 + -1x + x + 4x2 + -4x2
6 + x + -4x2 = -1x + x + 4x2 + -4x2

Combine like terms: -1x + x = 0
6 + x + -4x2 = 0 + 4x2 + -4x2
6 + x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
6 + x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-1.5 + -0.25x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.25x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.25x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.25x + x2 = 0 + 1.5
-0.25x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.25x + x2 = 1.5

The x term is x.  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.
-0.25x + 0.25 + x2 = 1.5 + 0.25

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

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + -0.25x + 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 = 0.822875656
x = 0.822875656

Simplifying
x = 0.822875656

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 = -1.822875656
x = -1.822875656

Simplifying
x = -1.822875656

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