3q^2=-27q^4+51q^3-9q^2

Solution for 3q^2=-27q^4+51q^3-9q^2 equation:

Simplifying
3q2 = -27q4 + 51q3 + -9q2

Reorder the terms:
3q2 = -9q2 + 51q3 + -27q4

Solving
3q2 = -9q2 + 51q3 + -27q4

Solving for variable 'q'.

Combine like terms: 3q2 + 9q2 = 12q2
12q2 + -51q3 + 27q4 = -9q2 + 51q3 + -27q4 + 9q2 + -51q3 + 27q4

Reorder the terms:
12q2 + -51q3 + 27q4 = -9q2 + 9q2 + 51q3 + -51q3 + -27q4 + 27q4

Combine like terms: -9q2 + 9q2 = 0
12q2 + -51q3 + 27q4 = 0 + 51q3 + -51q3 + -27q4 + 27q4
12q2 + -51q3 + 27q4 = 51q3 + -51q3 + -27q4 + 27q4

Combine like terms: 51q3 + -51q3 = 0
12q2 + -51q3 + 27q4 = 0 + -27q4 + 27q4
12q2 + -51q3 + 27q4 = -27q4 + 27q4

Combine like terms: -27q4 + 27q4 = 0
12q2 + -51q3 + 27q4 = 0

Factor out the Greatest Common Factor (GCF), '3q2'.
3q2(4 + -17q + 9q2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'q2' equal to zero and attempt to solve:

Simplifying
q2 = 0

Solving
q2 = 0

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

Simplifying
q2 = 0

Take the square root of each side:
q = {0}
Subproblem 2
Set the factor '(4 + -17q + 9q2)' equal to zero and attempt to solve:

Simplifying
4 + -17q + 9q2 = 0

Solving
4 + -17q + 9q2 = 0

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

Divide each side by '9'.
0.4444444444 + -1.888888889q + q2 = 0

Move the constant term to the right:

Add '-0.4444444444' to each side of the equation.
0.4444444444 + -1.888888889q + -0.4444444444 + q2 = 0 + -0.4444444444

Reorder the terms:
0.4444444444 + -0.4444444444 + -1.888888889q + q2 = 0 + -0.4444444444

Combine like terms: 0.4444444444 + -0.4444444444 = 0.0000000000
0.0000000000 + -1.888888889q + q2 = 0 + -0.4444444444
-1.888888889q + q2 = 0 + -0.4444444444

Combine like terms: 0 + -0.4444444444 = -0.4444444444
-1.888888889q + q2 = -0.4444444444

The q term is -1.888888889q.  Take half its coefficient (-0.9444444445).
Square it (0.8919753087) and add it to both sides.

Add '0.8919753087' to each side of the equation.
-1.888888889q + 0.8919753087 + q2 = -0.4444444444 + 0.8919753087

Reorder the terms:
0.8919753087 + -1.888888889q + q2 = -0.4444444444 + 0.8919753087

Combine like terms: -0.4444444444 + 0.8919753087 = 0.4475308643
0.8919753087 + -1.888888889q + q2 = 0.4475308643

Factor a perfect square on the left side:
(q + -0.9444444445)(q + -0.9444444445) = 0.4475308643

Calculate the square root of the right side: 0.668977477

Break this problem into two subproblems by setting 
(q + -0.9444444445) equal to 0.668977477 and -0.668977477.

Subproblem 1
q + -0.9444444445 = 0.668977477

Simplifying
q + -0.9444444445 = 0.668977477

Reorder the terms:
-0.9444444445 + q = 0.668977477

Solving
-0.9444444445 + q = 0.668977477

Solving for variable 'q'.

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

Add '0.9444444445' to each side of the equation.
-0.9444444445 + 0.9444444445 + q = 0.668977477 + 0.9444444445

Combine like terms: -0.9444444445 + 0.9444444445 = 0.0000000000
0.0000000000 + q = 0.668977477 + 0.9444444445
q = 0.668977477 + 0.9444444445

Combine like terms: 0.668977477 + 0.9444444445 = 1.6134219215
q = 1.6134219215

Simplifying
q = 1.6134219215

Subproblem 2
q + -0.9444444445 = -0.668977477

Simplifying
q + -0.9444444445 = -0.668977477

Reorder the terms:
-0.9444444445 + q = -0.668977477

Solving
-0.9444444445 + q = -0.668977477

Solving for variable 'q'.

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

Add '0.9444444445' to each side of the equation.
-0.9444444445 + 0.9444444445 + q = -0.668977477 + 0.9444444445

Combine like terms: -0.9444444445 + 0.9444444445 = 0.0000000000
0.0000000000 + q = -0.668977477 + 0.9444444445
q = -0.668977477 + 0.9444444445

Combine like terms: -0.668977477 + 0.9444444445 = 0.2754669675
q = 0.2754669675

Simplifying
q = 0.2754669675

Solution
The solution to the problem is based on the solutions
from the subproblems.
q = {1.6134219215, 0.2754669675}
Solution
q = {0, 1.6134219215, 0.2754669675}