0=y^2-4y-1

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Solution for 0=y^2-4y-1 equation:


Simplifying
0 = y2 + -4y + -1

Reorder the terms:
0 = -1 + -4y + y2

Solving
0 = -1 + -4y + y2

Solving for variable 'y'.

Combine like terms: 0 + 1 = 1
1 + 4y + -1y2 = -1 + -4y + y2 + 1 + 4y + -1y2

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

Combine like terms: -1 + 1 = 0
1 + 4y + -1y2 = 0 + -4y + 4y + y2 + -1y2
1 + 4y + -1y2 = -4y + 4y + y2 + -1y2

Combine like terms: -4y + 4y = 0
1 + 4y + -1y2 = 0 + y2 + -1y2
1 + 4y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
1 + 4y + -1y2 = 0

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

Divide each side by '-1'.
-1 + -4y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -4y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4y + y2 = 0 + 1
-4y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4y + y2 = 1

The y term is -4y.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

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

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

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

Factor a perfect square on the left side:
(y + -2)(y + -2) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(y + -2) equal to 2.236067978 and -2.236067978.

Subproblem 1

y + -2 = 2.236067978 Simplifying y + -2 = 2.236067978 Reorder the terms: -2 + y = 2.236067978 Solving -2 + y = 2.236067978 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = 2.236067978 + 2 Combine like terms: -2 + 2 = 0 0 + y = 2.236067978 + 2 y = 2.236067978 + 2 Combine like terms: 2.236067978 + 2 = 4.236067978 y = 4.236067978 Simplifying y = 4.236067978

Subproblem 2

y + -2 = -2.236067978 Simplifying y + -2 = -2.236067978 Reorder the terms: -2 + y = -2.236067978 Solving -2 + y = -2.236067978 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + y = -2.236067978 + 2 Combine like terms: -2 + 2 = 0 0 + y = -2.236067978 + 2 y = -2.236067978 + 2 Combine like terms: -2.236067978 + 2 = -0.236067978 y = -0.236067978 Simplifying y = -0.236067978

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.236067978, -0.236067978}

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