Y^2-6Y=3Y^2-12

Solution for Y^2-6Y=3Y^2-12 equation:

Simplifying
Y2 + -6Y = 3Y2 + -12

Reorder the terms:
-6Y + Y2 = 3Y2 + -12

Reorder the terms:
-6Y + Y2 = -12 + 3Y2

Solving
-6Y + Y2 = -12 + 3Y2

Solving for variable 'Y'.

Reorder the terms:
12 + -6Y + Y2 + -3Y2 = -12 + 3Y2 + 12 + -3Y2

Combine like terms: Y2 + -3Y2 = -2Y2
12 + -6Y + -2Y2 = -12 + 3Y2 + 12 + -3Y2

Reorder the terms:
12 + -6Y + -2Y2 = -12 + 12 + 3Y2 + -3Y2

Combine like terms: -12 + 12 = 0
12 + -6Y + -2Y2 = 0 + 3Y2 + -3Y2
12 + -6Y + -2Y2 = 3Y2 + -3Y2

Combine like terms: 3Y2 + -3Y2 = 0
12 + -6Y + -2Y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(6 + -3Y + -1Y2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(6 + -3Y + -1Y2)' equal to zero and attempt to solve:

Simplifying
6 + -3Y + -1Y2 = 0

Solving
6 + -3Y + -1Y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-6 + 3Y + Y2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3Y + 6 + Y2 = 0 + 6

Reorder the terms:
-6 + 6 + 3Y + Y2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 3Y + Y2 = 0 + 6
3Y + Y2 = 0 + 6

Combine like terms: 0 + 6 = 6
3Y + Y2 = 6

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 = 6 + 2.25

Reorder the terms:
2.25 + 3Y + Y2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + 3Y + Y2 = 8.25

Factor a perfect square on the left side:
(Y + 1.5)(Y + 1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(Y + 1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1
Y + 1.5 = 2.872281323

Simplifying
Y + 1.5 = 2.872281323

Reorder the terms:
1.5 + Y = 2.872281323

Solving
1.5 + Y = 2.872281323

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 = 2.872281323 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + Y = 2.872281323 + -1.5
Y = 2.872281323 + -1.5

Combine like terms: 2.872281323 + -1.5 = 1.372281323
Y = 1.372281323

Simplifying
Y = 1.372281323

Subproblem 2
Y + 1.5 = -2.872281323

Simplifying
Y + 1.5 = -2.872281323

Reorder the terms:
1.5 + Y = -2.872281323

Solving
1.5 + Y = -2.872281323

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 = -2.872281323 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + Y = -2.872281323 + -1.5
Y = -2.872281323 + -1.5

Combine like terms: -2.872281323 + -1.5 = -4.372281323
Y = -4.372281323

Simplifying
Y = -4.372281323

Solution
The solution to the problem is based on the solutions
from the subproblems.
Y = {1.372281323, -4.372281323}
Solution
Y = {1.372281323, -4.372281323}