3y^2-4y-1=y^2+2y+1

Solution for 3y^2-4y-1=y^2+2y+1 equation:

Simplifying
3y2 + -4y + -1 = y2 + 2y + 1

Reorder the terms:
-1 + -4y + 3y2 = y2 + 2y + 1

Reorder the terms:
-1 + -4y + 3y2 = 1 + 2y + y2

Solving
-1 + -4y + 3y2 = 1 + 2y + y2

Solving for variable 'y'.

Reorder the terms:
-1 + -1 + -4y + -2y + 3y2 + -1y2 = 1 + 2y + y2 + -1 + -2y + -1y2

Combine like terms: -1 + -1 = -2
-2 + -4y + -2y + 3y2 + -1y2 = 1 + 2y + y2 + -1 + -2y + -1y2

Combine like terms: -4y + -2y = -6y
-2 + -6y + 3y2 + -1y2 = 1 + 2y + y2 + -1 + -2y + -1y2

Combine like terms: 3y2 + -1y2 = 2y2
-2 + -6y + 2y2 = 1 + 2y + y2 + -1 + -2y + -1y2

Reorder the terms:
-2 + -6y + 2y2 = 1 + -1 + 2y + -2y + y2 + -1y2

Combine like terms: 1 + -1 = 0
-2 + -6y + 2y2 = 0 + 2y + -2y + y2 + -1y2
-2 + -6y + 2y2 = 2y + -2y + y2 + -1y2

Combine like terms: 2y + -2y = 0
-2 + -6y + 2y2 = 0 + y2 + -1y2
-2 + -6y + 2y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-2 + -6y + 2y2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + -3y + y2) = 0

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

Simplifying
-1 + -3y + y2 = 0

Solving
-1 + -3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -3y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -3y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -3y + y2 = 0 + 1
-3y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-3y + y2 = 1

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 = 1 + 2.25

Reorder the terms:
2.25 + -3y + y2 = 1 + 2.25

Combine like terms: 1 + 2.25 = 3.25
2.25 + -3y + y2 = 3.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(y + -1.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
y + -1.5 = 1.802775638

Simplifying
y + -1.5 = 1.802775638

Reorder the terms:
-1.5 + y = 1.802775638

Solving
-1.5 + y = 1.802775638

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 = 1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = 1.802775638 + 1.5
y = 1.802775638 + 1.5

Combine like terms: 1.802775638 + 1.5 = 3.302775638
y = 3.302775638

Simplifying
y = 3.302775638

Subproblem 2
y + -1.5 = -1.802775638

Simplifying
y + -1.5 = -1.802775638

Reorder the terms:
-1.5 + y = -1.802775638

Solving
-1.5 + y = -1.802775638

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 = -1.802775638 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = -1.802775638 + 1.5
y = -1.802775638 + 1.5

Combine like terms: -1.802775638 + 1.5 = -0.302775638
y = -0.302775638

Simplifying
y = -0.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.302775638, -0.302775638}
Solution
y = {3.302775638, -0.302775638}