-4q^2+q+6=0

Solution for -4q^2+q+6=0 equation:

Simplifying
-4q2 + q + 6 = 0

Reorder the terms:
6 + q + -4q2 = 0

Solving
6 + q + -4q2 = 0

Solving for variable 'q'.

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

Divide each side by '-4'.
-1.5 + -0.25q + q2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.25q + 1.5 + q2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.25q + q2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.25q + q2 = 0 + 1.5
-0.25q + q2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.25q + q2 = 1.5

The q term is q.  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.
-0.25q + 0.25 + q2 = 1.5 + 0.25

Reorder the terms:
0.25 + -0.25q + q2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + -0.25q + q2 = 1.75

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

Calculate the square root of the right side: 1.322875656

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

Subproblem 1
q + 0.5 = 1.322875656

Simplifying
q + 0.5 = 1.322875656

Reorder the terms:
0.5 + q = 1.322875656

Solving
0.5 + q = 1.322875656

Solving for variable 'q'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + q = 1.322875656 + -0.5

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

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

Simplifying
q = 0.822875656

Subproblem 2
q + 0.5 = -1.322875656

Simplifying
q + 0.5 = -1.322875656

Reorder the terms:
0.5 + q = -1.322875656

Solving
0.5 + q = -1.322875656

Solving for variable 'q'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + q = -1.322875656 + -0.5

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

Combine like terms: -1.322875656 + -0.5 = -1.822875656
q = -1.822875656

Simplifying
q = -1.822875656

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