177y^2-15y-6=0

Solution for 177y^2-15y-6=0 equation:

Simplifying
177y2 + -15y + -6 = 0

Reorder the terms:
-6 + -15y + 177y2 = 0

Solving
-6 + -15y + 177y2 = 0

Solving for variable 'y'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-2 + -5y + 59y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-2 + -5y + 59y2)' equal to zero and attempt to solve:

Simplifying
-2 + -5y + 59y2 = 0

Solving
-2 + -5y + 59y2 = 0

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

Divide each side by '59'.
-0.03389830508 + -0.08474576271y + y2 = 0

Move the constant term to the right:

Add '0.03389830508' to each side of the equation.
-0.03389830508 + -0.08474576271y + 0.03389830508 + y2 = 0 + 0.03389830508

Reorder the terms:
-0.03389830508 + 0.03389830508 + -0.08474576271y + y2 = 0 + 0.03389830508

Combine like terms: -0.03389830508 + 0.03389830508 = 0.00000000000
0.00000000000 + -0.08474576271y + y2 = 0 + 0.03389830508
-0.08474576271y + y2 = 0 + 0.03389830508

Combine like terms: 0 + 0.03389830508 = 0.03389830508
-0.08474576271y + y2 = 0.03389830508

The y term is -0.08474576271y.  Take half its coefficient (-0.04237288136).
Square it (0.001795461075) and add it to both sides.

Add '0.001795461075' to each side of the equation.
-0.08474576271y + 0.001795461075 + y2 = 0.03389830508 + 0.001795461075

Reorder the terms:
0.001795461075 + -0.08474576271y + y2 = 0.03389830508 + 0.001795461075

Combine like terms: 0.03389830508 + 0.001795461075 = 0.035693766155
0.001795461075 + -0.08474576271y + y2 = 0.035693766155

Factor a perfect square on the left side:
(y + -0.04237288136)(y + -0.04237288136) = 0.035693766155

Calculate the square root of the right side: 0.188927939

Break this problem into two subproblems by setting 
(y + -0.04237288136) equal to 0.188927939 and -0.188927939.

Subproblem 1
y + -0.04237288136 = 0.188927939

Simplifying
y + -0.04237288136 = 0.188927939

Reorder the terms:
-0.04237288136 + y = 0.188927939

Solving
-0.04237288136 + y = 0.188927939

Solving for variable 'y'.

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

Add '0.04237288136' to each side of the equation.
-0.04237288136 + 0.04237288136 + y = 0.188927939 + 0.04237288136

Combine like terms: -0.04237288136 + 0.04237288136 = 0.00000000000
0.00000000000 + y = 0.188927939 + 0.04237288136
y = 0.188927939 + 0.04237288136

Combine like terms: 0.188927939 + 0.04237288136 = 0.23130082036
y = 0.23130082036

Simplifying
y = 0.23130082036

Subproblem 2
y + -0.04237288136 = -0.188927939

Simplifying
y + -0.04237288136 = -0.188927939

Reorder the terms:
-0.04237288136 + y = -0.188927939

Solving
-0.04237288136 + y = -0.188927939

Solving for variable 'y'.

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

Add '0.04237288136' to each side of the equation.
-0.04237288136 + 0.04237288136 + y = -0.188927939 + 0.04237288136

Combine like terms: -0.04237288136 + 0.04237288136 = 0.00000000000
0.00000000000 + y = -0.188927939 + 0.04237288136
y = -0.188927939 + 0.04237288136

Combine like terms: -0.188927939 + 0.04237288136 = -0.14655505764
y = -0.14655505764

Simplifying
y = -0.14655505764

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.23130082036, -0.14655505764}
Solution
y = {0.23130082036, -0.14655505764}