6y^2-21y-291=0

Solution for 6y^2-21y-291=0 equation:

Simplifying
6y2 + -21y + -291 = 0

Reorder the terms:
-291 + -21y + 6y2 = 0

Solving
-291 + -21y + 6y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-97 + -7y + 2y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-97 + -7y + 2y2)' equal to zero and attempt to solve:

Simplifying
-97 + -7y + 2y2 = 0

Solving
-97 + -7y + 2y2 = 0

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

Divide each side by '2'.
-48.5 + -3.5y + y2 = 0

Move the constant term to the right:

Add '48.5' to each side of the equation.
-48.5 + -3.5y + 48.5 + y2 = 0 + 48.5

Reorder the terms:
-48.5 + 48.5 + -3.5y + y2 = 0 + 48.5

Combine like terms: -48.5 + 48.5 = 0.0
0.0 + -3.5y + y2 = 0 + 48.5
-3.5y + y2 = 0 + 48.5

Combine like terms: 0 + 48.5 = 48.5
-3.5y + y2 = 48.5

The y term is -3.5y.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5y + 3.0625 + y2 = 48.5 + 3.0625

Reorder the terms:
3.0625 + -3.5y + y2 = 48.5 + 3.0625

Combine like terms: 48.5 + 3.0625 = 51.5625
3.0625 + -3.5y + y2 = 51.5625

Factor a perfect square on the left side:
(y + -1.75)(y + -1.75) = 51.5625

Calculate the square root of the right side: 7.180703308

Break this problem into two subproblems by setting 
(y + -1.75) equal to 7.180703308 and -7.180703308.

Subproblem 1
y + -1.75 = 7.180703308

Simplifying
y + -1.75 = 7.180703308

Reorder the terms:
-1.75 + y = 7.180703308

Solving
-1.75 + y = 7.180703308

Solving for variable 'y'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + y = 7.180703308 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = 7.180703308 + 1.75
y = 7.180703308 + 1.75

Combine like terms: 7.180703308 + 1.75 = 8.930703308
y = 8.930703308

Simplifying
y = 8.930703308

Subproblem 2
y + -1.75 = -7.180703308

Simplifying
y + -1.75 = -7.180703308

Reorder the terms:
-1.75 + y = -7.180703308

Solving
-1.75 + y = -7.180703308

Solving for variable 'y'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + y = -7.180703308 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = -7.180703308 + 1.75
y = -7.180703308 + 1.75

Combine like terms: -7.180703308 + 1.75 = -5.430703308
y = -5.430703308

Simplifying
y = -5.430703308

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {8.930703308, -5.430703308}
Solution
y = {8.930703308, -5.430703308}