6y(y-4)+3(y+3)=57

Solution for 6y(y-4)+3(y+3)=57 equation:

Simplifying
6y(y + -4) + 3(y + 3) = 57

Reorder the terms:
6y(-4 + y) + 3(y + 3) = 57
(-4 * 6y + y * 6y) + 3(y + 3) = 57
(-24y + 6y2) + 3(y + 3) = 57

Reorder the terms:
-24y + 6y2 + 3(3 + y) = 57
-24y + 6y2 + (3 * 3 + y * 3) = 57
-24y + 6y2 + (9 + 3y) = 57

Reorder the terms:
9 + -24y + 3y + 6y2 = 57

Combine like terms: -24y + 3y = -21y
9 + -21y + 6y2 = 57

Solving
9 + -21y + 6y2 = 57

Solving for variable 'y'.

Reorder the terms:
9 + -57 + -21y + 6y2 = 57 + -57

Combine like terms: 9 + -57 = -48
-48 + -21y + 6y2 = 57 + -57

Combine like terms: 57 + -57 = 0
-48 + -21y + 6y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-16 + -7y + 2y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-16 + -7y + 2y2)' equal to zero and attempt to solve:

Simplifying
-16 + -7y + 2y2 = 0

Solving
-16 + -7y + 2y2 = 0

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

Divide each side by '2'.
-8 + -3.5y + y2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -3.5y + 8 + y2 = 0 + 8

Reorder the terms:
-8 + 8 + -3.5y + y2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -3.5y + y2 = 0 + 8
-3.5y + y2 = 0 + 8

Combine like terms: 0 + 8 = 8
-3.5y + y2 = 8

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 = 8 + 3.0625

Reorder the terms:
3.0625 + -3.5y + y2 = 8 + 3.0625

Combine like terms: 8 + 3.0625 = 11.0625
3.0625 + -3.5y + y2 = 11.0625

Factor a perfect square on the left side:
(y + -1.75)(y + -1.75) = 11.0625

Calculate the square root of the right side: 3.326033674

Break this problem into two subproblems by setting 
(y + -1.75) equal to 3.326033674 and -3.326033674.

Subproblem 1
y + -1.75 = 3.326033674

Simplifying
y + -1.75 = 3.326033674

Reorder the terms:
-1.75 + y = 3.326033674

Solving
-1.75 + y = 3.326033674

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 = 3.326033674 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = 3.326033674 + 1.75
y = 3.326033674 + 1.75

Combine like terms: 3.326033674 + 1.75 = 5.076033674
y = 5.076033674

Simplifying
y = 5.076033674

Subproblem 2
y + -1.75 = -3.326033674

Simplifying
y + -1.75 = -3.326033674

Reorder the terms:
-1.75 + y = -3.326033674

Solving
-1.75 + y = -3.326033674

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 = -3.326033674 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + y = -3.326033674 + 1.75
y = -3.326033674 + 1.75

Combine like terms: -3.326033674 + 1.75 = -1.576033674
y = -1.576033674

Simplifying
y = -1.576033674

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {5.076033674, -1.576033674}
Solution
y = {5.076033674, -1.576033674}