2y(y-3)=7-1

Solution for 2y(y-3)=7-1 equation:

Simplifying
2y(y + -3) = 7 + -1

Reorder the terms:
2y(-3 + y) = 7 + -1
(-3 * 2y + y * 2y) = 7 + -1
(-6y + 2y2) = 7 + -1

Combine like terms: 7 + -1 = 6
-6y + 2y2 = 6

Solving
-6y + 2y2 = 6

Solving for variable 'y'.

Reorder the terms:
-6 + -6y + 2y2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + -6y + 2y2 = 0

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

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

Simplifying
-3 + -3y + y2 = 0

Solving
-3 + -3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -3y + 3 + y2 = 0 + 3

Reorder the terms:
-3 + 3 + -3y + y2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -3y + y2 = 0 + 3
-3y + y2 = 0 + 3

Combine like terms: 0 + 3 = 3
-3y + y2 = 3

The y term is -3y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3y + 2.25 + y2 = 3 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 3 + 2.25

Combine like terms: 3 + 2.25 = 5.25
2.25 + -3y + y2 = 5.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(y + -1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
y + -1.5 = 2.291287847

Simplifying
y + -1.5 = 2.291287847

Reorder the terms:
-1.5 + y = 2.291287847

Solving
-1.5 + y = 2.291287847

Solving for variable 'y'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = 2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = 2.291287847 + 1.5
y = 2.291287847 + 1.5

Combine like terms: 2.291287847 + 1.5 = 3.791287847
y = 3.791287847

Simplifying
y = 3.791287847

Subproblem 2
y + -1.5 = -2.291287847

Simplifying
y + -1.5 = -2.291287847

Reorder the terms:
-1.5 + y = -2.291287847

Solving
-1.5 + y = -2.291287847

Solving for variable 'y'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = -2.291287847 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = -2.291287847 + 1.5
y = -2.291287847 + 1.5

Combine like terms: -2.291287847 + 1.5 = -0.791287847
y = -0.791287847

Simplifying
y = -0.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.791287847, -0.791287847}
Solution
y = {3.791287847, -0.791287847}