5y^2=-9y-1

Solution for 5y^2=-9y-1 equation:

Simplifying
5y2 = -9y + -1

Reorder the terms:
5y2 = -1 + -9y

Solving
5y2 = -1 + -9y

Solving for variable 'y'.

Reorder the terms:
1 + 9y + 5y2 = -1 + -9y + 1 + 9y

Reorder the terms:
1 + 9y + 5y2 = -1 + 1 + -9y + 9y

Combine like terms: -1 + 1 = 0
1 + 9y + 5y2 = 0 + -9y + 9y
1 + 9y + 5y2 = -9y + 9y

Combine like terms: -9y + 9y = 0
1 + 9y + 5y2 = 0

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

Divide each side by '5'.
0.2 + 1.8y + y2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 1.8y + -0.2 + y2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 1.8y + y2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 1.8y + y2 = 0 + -0.2
1.8y + y2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
1.8y + y2 = -0.2

The y term is 1.8y.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8y + 0.81 + y2 = -0.2 + 0.81

Reorder the terms:
0.81 + 1.8y + y2 = -0.2 + 0.81

Combine like terms: -0.2 + 0.81 = 0.61
0.81 + 1.8y + y2 = 0.61

Factor a perfect square on the left side:
(y + 0.9)(y + 0.9) = 0.61

Calculate the square root of the right side: 0.781024968

Break this problem into two subproblems by setting 
(y + 0.9) equal to 0.781024968 and -0.781024968.

Subproblem 1
y + 0.9 = 0.781024968

Simplifying
y + 0.9 = 0.781024968

Reorder the terms:
0.9 + y = 0.781024968

Solving
0.9 + y = 0.781024968

Solving for variable 'y'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + y = 0.781024968 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + y = 0.781024968 + -0.9
y = 0.781024968 + -0.9

Combine like terms: 0.781024968 + -0.9 = -0.118975032
y = -0.118975032

Simplifying
y = -0.118975032

Subproblem 2
y + 0.9 = -0.781024968

Simplifying
y + 0.9 = -0.781024968

Reorder the terms:
0.9 + y = -0.781024968

Solving
0.9 + y = -0.781024968

Solving for variable 'y'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + y = -0.781024968 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + y = -0.781024968 + -0.9
y = -0.781024968 + -0.9

Combine like terms: -0.781024968 + -0.9 = -1.681024968
y = -1.681024968

Simplifying
y = -1.681024968

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.118975032, -1.681024968}