6y^2-9y=1

Solution for 6y^2-9y=1 equation:

Simplifying
6y2 + -9y = 1

Reorder the terms:
-9y + 6y2 = 1

Solving
-9y + 6y2 = 1

Solving for variable 'y'.

Reorder the terms:
-1 + -9y + 6y2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -9y + 6y2 = 0

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

Divide each side by '6'.
-0.1666666667 + -1.5y + y2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.5y + 0.1666666667 + y2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.5y + y2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.5y + y2 = 0 + 0.1666666667
-1.5y + y2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.5y + y2 = 0.1666666667

The y term is -1.5y.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5y + 0.5625 + y2 = 0.1666666667 + 0.5625

Reorder the terms:
0.5625 + -1.5y + y2 = 0.1666666667 + 0.5625

Combine like terms: 0.1666666667 + 0.5625 = 0.7291666667
0.5625 + -1.5y + y2 = 0.7291666667

Factor a perfect square on the left side:
(y + -0.75)(y + -0.75) = 0.7291666667

Calculate the square root of the right side: 0.853912564

Break this problem into two subproblems by setting 
(y + -0.75) equal to 0.853912564 and -0.853912564.

Subproblem 1
y + -0.75 = 0.853912564

Simplifying
y + -0.75 = 0.853912564

Reorder the terms:
-0.75 + y = 0.853912564

Solving
-0.75 + y = 0.853912564

Solving for variable 'y'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + y = 0.853912564 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + y = 0.853912564 + 0.75
y = 0.853912564 + 0.75

Combine like terms: 0.853912564 + 0.75 = 1.603912564
y = 1.603912564

Simplifying
y = 1.603912564

Subproblem 2
y + -0.75 = -0.853912564

Simplifying
y + -0.75 = -0.853912564

Reorder the terms:
-0.75 + y = -0.853912564

Solving
-0.75 + y = -0.853912564

Solving for variable 'y'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + y = -0.853912564 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + y = -0.853912564 + 0.75
y = -0.853912564 + 0.75

Combine like terms: -0.853912564 + 0.75 = -0.103912564
y = -0.103912564

Simplifying
y = -0.103912564

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.603912564, -0.103912564}