y^2-7y-1=0

Solution for y^2-7y-1=0 equation:

Simplifying
y2 + -7y + -1 = 0

Reorder the terms:
-1 + -7y + y2 = 0

Solving
-1 + -7y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -7y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -7y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -7y + y2 = 0 + 1
-7y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-7y + y2 = 1

The y term is -7y.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7y + 12.25 + y2 = 1 + 12.25

Reorder the terms:
12.25 + -7y + y2 = 1 + 12.25

Combine like terms: 1 + 12.25 = 13.25
12.25 + -7y + y2 = 13.25

Factor a perfect square on the left side:
(y + -3.5)(y + -3.5) = 13.25

Calculate the square root of the right side: 3.640054945

Break this problem into two subproblems by setting 
(y + -3.5) equal to 3.640054945 and -3.640054945.

Subproblem 1
y + -3.5 = 3.640054945

Simplifying
y + -3.5 = 3.640054945

Reorder the terms:
-3.5 + y = 3.640054945

Solving
-3.5 + y = 3.640054945

Solving for variable 'y'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + y = 3.640054945 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + y = 3.640054945 + 3.5
y = 3.640054945 + 3.5

Combine like terms: 3.640054945 + 3.5 = 7.140054945
y = 7.140054945

Simplifying
y = 7.140054945

Subproblem 2
y + -3.5 = -3.640054945

Simplifying
y + -3.5 = -3.640054945

Reorder the terms:
-3.5 + y = -3.640054945

Solving
-3.5 + y = -3.640054945

Solving for variable 'y'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + y = -3.640054945 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + y = -3.640054945 + 3.5
y = -3.640054945 + 3.5

Combine like terms: -3.640054945 + 3.5 = -0.140054945
y = -0.140054945

Simplifying
y = -0.140054945

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {7.140054945, -0.140054945}