y^2-3y=-1

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

Simplifying
y2 + -3y = -1

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

Solving
-3y + y2 = -1

Solving for variable 'y'.

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

Combine like terms: -1 + 1 = 0
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 = 1.25
2.25 + -3y + y2 = 1.25

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(y + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
y + -1.5 = 1.118033989

Simplifying
y + -1.5 = 1.118033989

Reorder the terms:
-1.5 + y = 1.118033989

Solving
-1.5 + y = 1.118033989

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 = 1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = 1.118033989 + 1.5
y = 1.118033989 + 1.5

Combine like terms: 1.118033989 + 1.5 = 2.618033989
y = 2.618033989

Simplifying
y = 2.618033989

Subproblem 2
y + -1.5 = -1.118033989

Simplifying
y + -1.5 = -1.118033989

Reorder the terms:
-1.5 + y = -1.118033989

Solving
-1.5 + y = -1.118033989

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 = -1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = -1.118033989 + 1.5
y = -1.118033989 + 1.5

Combine like terms: -1.118033989 + 1.5 = 0.381966011
y = 0.381966011

Simplifying
y = 0.381966011

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.618033989, 0.381966011}