3y(2y-4)=21

Solution for 3y(2y-4)=21 equation:

Simplifying
3y(2y + -4) = 21

Reorder the terms:
3y(-4 + 2y) = 21
(-4 * 3y + 2y * 3y) = 21
(-12y + 6y2) = 21

Solving
-12y + 6y2 = 21

Solving for variable 'y'.

Reorder the terms:
-21 + -12y + 6y2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + -12y + 6y2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + -4y + 2y2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-7 + -4y + 2y2)' equal to zero and attempt to solve:

Simplifying
-7 + -4y + 2y2 = 0

Solving
-7 + -4y + 2y2 = 0

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

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

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -2y + 3.5 + y2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -2y + y2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -2y + y2 = 0 + 3.5
-2y + y2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-2y + y2 = 3.5

The y term is -2y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2y + 1 + y2 = 3.5 + 1

Reorder the terms:
1 + -2y + y2 = 3.5 + 1

Combine like terms: 3.5 + 1 = 4.5
1 + -2y + y2 = 4.5

Factor a perfect square on the left side:
(y + -1)(y + -1) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(y + -1) equal to 2.121320344 and -2.121320344.

Subproblem 1
y + -1 = 2.121320344

Simplifying
y + -1 = 2.121320344

Reorder the terms:
-1 + y = 2.121320344

Solving
-1 + y = 2.121320344

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = 2.121320344 + 1

Combine like terms: -1 + 1 = 0
0 + y = 2.121320344 + 1
y = 2.121320344 + 1

Combine like terms: 2.121320344 + 1 = 3.121320344
y = 3.121320344

Simplifying
y = 3.121320344

Subproblem 2
y + -1 = -2.121320344

Simplifying
y + -1 = -2.121320344

Reorder the terms:
-1 + y = -2.121320344

Solving
-1 + y = -2.121320344

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = -2.121320344 + 1

Combine like terms: -1 + 1 = 0
0 + y = -2.121320344 + 1
y = -2.121320344 + 1

Combine like terms: -2.121320344 + 1 = -1.121320344
y = -1.121320344

Simplifying
y = -1.121320344

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.121320344, -1.121320344}
Solution
y = {3.121320344, -1.121320344}