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 = 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}