4q(8q+9)=7-(12q-9)

Solution for 4q(8q+9)=7-(12q-9) equation:

Simplifying
4q(8q + 9) = 7 + -1(12q + -9)

Reorder the terms:
4q(9 + 8q) = 7 + -1(12q + -9)
(9 * 4q + 8q * 4q) = 7 + -1(12q + -9)
(36q + 32q2) = 7 + -1(12q + -9)

Reorder the terms:
36q + 32q2 = 7 + -1(-9 + 12q)
36q + 32q2 = 7 + (-9 * -1 + 12q * -1)
36q + 32q2 = 7 + (9 + -12q)

Combine like terms: 7 + 9 = 16
36q + 32q2 = 16 + -12q

Solving
36q + 32q2 = 16 + -12q

Solving for variable 'q'.

Reorder the terms:
-16 + 36q + 12q + 32q2 = 16 + -12q + -16 + 12q

Combine like terms: 36q + 12q = 48q
-16 + 48q + 32q2 = 16 + -12q + -16 + 12q

Reorder the terms:
-16 + 48q + 32q2 = 16 + -16 + -12q + 12q

Combine like terms: 16 + -16 = 0
-16 + 48q + 32q2 = 0 + -12q + 12q
-16 + 48q + 32q2 = -12q + 12q

Combine like terms: -12q + 12q = 0
-16 + 48q + 32q2 = 0

Factor out the Greatest Common Factor (GCF), '16'.
16(-1 + 3q + 2q2) = 0

Ignore the factor 16.
Subproblem 1
Set the factor '(-1 + 3q + 2q2)' equal to zero and attempt to solve:

Simplifying
-1 + 3q + 2q2 = 0

Solving
-1 + 3q + 2q2 = 0

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

Divide each side by '2'.
-0.5 + 1.5q + q2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 1.5q + 0.5 + q2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 1.5q + q2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 1.5q + q2 = 0 + 0.5
1.5q + q2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
1.5q + q2 = 0.5

The q term is 1.5q.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5q + 0.5625 + q2 = 0.5 + 0.5625

Reorder the terms:
0.5625 + 1.5q + q2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + 1.5q + q2 = 1.0625

Factor a perfect square on the left side:
(q + 0.75)(q + 0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(q + 0.75) equal to 1.030776406 and -1.030776406.

Subproblem 1
q + 0.75 = 1.030776406

Simplifying
q + 0.75 = 1.030776406

Reorder the terms:
0.75 + q = 1.030776406

Solving
0.75 + q = 1.030776406

Solving for variable 'q'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + q = 1.030776406 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + q = 1.030776406 + -0.75
q = 1.030776406 + -0.75

Combine like terms: 1.030776406 + -0.75 = 0.280776406
q = 0.280776406

Simplifying
q = 0.280776406

Subproblem 2
q + 0.75 = -1.030776406

Simplifying
q + 0.75 = -1.030776406

Reorder the terms:
0.75 + q = -1.030776406

Solving
0.75 + q = -1.030776406

Solving for variable 'q'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + q = -1.030776406 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + q = -1.030776406 + -0.75
q = -1.030776406 + -0.75

Combine like terms: -1.030776406 + -0.75 = -1.780776406
q = -1.780776406

Simplifying
q = -1.780776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {0.280776406, -1.780776406}
Solution
q = {0.280776406, -1.780776406}