4q^2+3q=3q-4q+18

Solution for 4q^2+3q=3q-4q+18 equation:

Simplifying
4q2 + 3q = 3q + -4q + 18

Reorder the terms:
3q + 4q2 = 3q + -4q + 18

Reorder the terms:
3q + 4q2 = 18 + 3q + -4q

Combine like terms: 3q + -4q = -1q
3q + 4q2 = 18 + -1q

Solving
3q + 4q2 = 18 + -1q

Solving for variable 'q'.

Reorder the terms:
-18 + 3q + q + 4q2 = 18 + -1q + -18 + q

Combine like terms: 3q + q = 4q
-18 + 4q + 4q2 = 18 + -1q + -18 + q

Reorder the terms:
-18 + 4q + 4q2 = 18 + -18 + -1q + q

Combine like terms: 18 + -18 = 0
-18 + 4q + 4q2 = 0 + -1q + q
-18 + 4q + 4q2 = -1q + q

Combine like terms: -1q + q = 0
-18 + 4q + 4q2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-9 + 2q + 2q2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-9 + 2q + 2q2)' equal to zero and attempt to solve:

Simplifying
-9 + 2q + 2q2 = 0

Solving
-9 + 2q + 2q2 = 0

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

Divide each side by '2'.
-4.5 + q + q2 = 0

Move the constant term to the right:

Add '4.5' to each side of the equation.
-4.5 + q + 4.5 + q2 = 0 + 4.5

Reorder the terms:
-4.5 + 4.5 + q + q2 = 0 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + q + q2 = 0 + 4.5
q + q2 = 0 + 4.5

Combine like terms: 0 + 4.5 = 4.5
q + q2 = 4.5

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.
 + 0.25 + q2 = 4.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + q2 = 4.5 + 0.25

Combine like terms: 4.5 + 0.25 = 4.75
1.25 + q2 = 4.75

Factor a perfect square on the left side:
(q + 0.5)(q + 0.5) = 4.75

Calculate the square root of the right side: 2.179449472

Break this problem into two subproblems by setting 
(q + 0.5) equal to 2.179449472 and -2.179449472.

Subproblem 1
q + 0.5 = 2.179449472

Simplifying
q + 0.5 = 2.179449472

Reorder the terms:
0.5 + q = 2.179449472

Solving
0.5 + q = 2.179449472

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 = 2.179449472 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + q = 2.179449472 + -0.5
q = 2.179449472 + -0.5

Combine like terms: 2.179449472 + -0.5 = 1.679449472
q = 1.679449472

Simplifying
q = 1.679449472

Subproblem 2
q + 0.5 = -2.179449472

Simplifying
q + 0.5 = -2.179449472

Reorder the terms:
0.5 + q = -2.179449472

Solving
0.5 + q = -2.179449472

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 = -2.179449472 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + q = -2.179449472 + -0.5
q = -2.179449472 + -0.5

Combine like terms: -2.179449472 + -0.5 = -2.679449472
q = -2.679449472

Simplifying
q = -2.679449472

Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {1.679449472, -2.679449472}
Solution
q = {1.679449472, -2.679449472}