8y^2=-9y+4

Solution for 8y^2=-9y+4 equation:

Simplifying
8y2 = -9y + 4

Reorder the terms:
8y2 = 4 + -9y

Solving
8y2 = 4 + -9y

Solving for variable 'y'.

Reorder the terms:
-4 + 9y + 8y2 = 4 + -9y + -4 + 9y

Reorder the terms:
-4 + 9y + 8y2 = 4 + -4 + -9y + 9y

Combine like terms: 4 + -4 = 0
-4 + 9y + 8y2 = 0 + -9y + 9y
-4 + 9y + 8y2 = -9y + 9y

Combine like terms: -9y + 9y = 0
-4 + 9y + 8y2 = 0

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

Divide each side by '8'.
-0.5 + 1.125y + y2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 1.125y + 0.5 + y2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 1.125y + y2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 1.125y + y2 = 0 + 0.5
1.125y + y2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
1.125y + y2 = 0.5

The y term is 1.125y.  Take half its coefficient (0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
1.125y + 0.31640625 + y2 = 0.5 + 0.31640625

Reorder the terms:
0.31640625 + 1.125y + y2 = 0.5 + 0.31640625

Combine like terms: 0.5 + 0.31640625 = 0.81640625
0.31640625 + 1.125y + y2 = 0.81640625

Factor a perfect square on the left side:
(y + 0.5625)(y + 0.5625) = 0.81640625

Calculate the square root of the right side: 0.903552018

Break this problem into two subproblems by setting 
(y + 0.5625) equal to 0.903552018 and -0.903552018.

Subproblem 1
y + 0.5625 = 0.903552018

Simplifying
y + 0.5625 = 0.903552018

Reorder the terms:
0.5625 + y = 0.903552018

Solving
0.5625 + y = 0.903552018

Solving for variable 'y'.

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

Add '-0.5625' to each side of the equation.
0.5625 + -0.5625 + y = 0.903552018 + -0.5625

Combine like terms: 0.5625 + -0.5625 = 0.0000
0.0000 + y = 0.903552018 + -0.5625
y = 0.903552018 + -0.5625

Combine like terms: 0.903552018 + -0.5625 = 0.341052018
y = 0.341052018

Simplifying
y = 0.341052018

Subproblem 2
y + 0.5625 = -0.903552018

Simplifying
y + 0.5625 = -0.903552018

Reorder the terms:
0.5625 + y = -0.903552018

Solving
0.5625 + y = -0.903552018

Solving for variable 'y'.

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

Add '-0.5625' to each side of the equation.
0.5625 + -0.5625 + y = -0.903552018 + -0.5625

Combine like terms: 0.5625 + -0.5625 = 0.0000
0.0000 + y = -0.903552018 + -0.5625
y = -0.903552018 + -0.5625

Combine like terms: -0.903552018 + -0.5625 = -1.466052018
y = -1.466052018

Simplifying
y = -1.466052018

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.341052018, -1.466052018}