5y^2-y=1

Solution for 5y^2-y=1 equation:

Simplifying
5y2 + -1y = 1

Reorder the terms:
-1y + 5y2 = 1

Solving
-1y + 5y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + -1y + 5y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -1y + 5y2 = 0

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

Divide each side by '5'.
-0.2 + -0.2y + y2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -0.2y + 0.2 + y2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -0.2y + y2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -0.2y + y2 = 0 + 0.2
-0.2y + y2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-0.2y + y2 = 0.2

The y term is -0.2y.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2y + 0.01 + y2 = 0.2 + 0.01

Reorder the terms:
0.01 + -0.2y + y2 = 0.2 + 0.01

Combine like terms: 0.2 + 0.01 = 0.21
0.01 + -0.2y + y2 = 0.21

Factor a perfect square on the left side:
(y + -0.1)(y + -0.1) = 0.21

Calculate the square root of the right side: 0.458257569

Break this problem into two subproblems by setting 
(y + -0.1) equal to 0.458257569 and -0.458257569.

Subproblem 1
y + -0.1 = 0.458257569

Simplifying
y + -0.1 = 0.458257569

Reorder the terms:
-0.1 + y = 0.458257569

Solving
-0.1 + y = 0.458257569

Solving for variable 'y'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + y = 0.458257569 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + y = 0.458257569 + 0.1
y = 0.458257569 + 0.1

Combine like terms: 0.458257569 + 0.1 = 0.558257569
y = 0.558257569

Simplifying
y = 0.558257569

Subproblem 2
y + -0.1 = -0.458257569

Simplifying
y + -0.1 = -0.458257569

Reorder the terms:
-0.1 + y = -0.458257569

Solving
-0.1 + y = -0.458257569

Solving for variable 'y'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + y = -0.458257569 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + y = -0.458257569 + 0.1
y = -0.458257569 + 0.1

Combine like terms: -0.458257569 + 0.1 = -0.358257569
y = -0.358257569

Simplifying
y = -0.358257569

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.558257569, -0.358257569}