y^2-2y-1=3

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

Simplifying
y2 + -2y + -1 = 3

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

Solving
-1 + -2y + y2 = 3

Solving for variable 'y'.

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

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

Combine like terms: 3 + -3 = 0
-4 + -2y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + -2y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2y + y2 = 0 + 4
-2y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2y + y2 = 4

The y term is -2y.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2y + 1 + y2 = 4 + 1

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

Combine like terms: 4 + 1 = 5
1 + -2y + y2 = 5

Factor a perfect square on the left side:
(y + -1)(y + -1) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(y + -1) equal to 2.236067978 and -2.236067978.

Subproblem 1
y + -1 = 2.236067978

Simplifying
y + -1 = 2.236067978

Reorder the terms:
-1 + y = 2.236067978

Solving
-1 + y = 2.236067978

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = 2.236067978 + 1

Combine like terms: -1 + 1 = 0
0 + y = 2.236067978 + 1
y = 2.236067978 + 1

Combine like terms: 2.236067978 + 1 = 3.236067978
y = 3.236067978

Simplifying
y = 3.236067978

Subproblem 2
y + -1 = -2.236067978

Simplifying
y + -1 = -2.236067978

Reorder the terms:
-1 + y = -2.236067978

Solving
-1 + y = -2.236067978

Solving for variable 'y'.

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

Add '1' to each side of the equation.
-1 + 1 + y = -2.236067978 + 1

Combine like terms: -1 + 1 = 0
0 + y = -2.236067978 + 1
y = -2.236067978 + 1

Combine like terms: -2.236067978 + 1 = -1.236067978
y = -1.236067978

Simplifying
y = -1.236067978

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {3.236067978, -1.236067978}