y^2+4.5y-9=0

Solution for y^2+4.5y-9=0 equation:

Simplifying
y2 + 4.5y + -9 = 0

Reorder the terms:
-9 + 4.5y + y2 = 0

Solving
-9 + 4.5y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 4.5y + 9 + y2 = 0 + 9

Reorder the terms:
-9 + 9 + 4.5y + y2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 4.5y + y2 = 0 + 9
4.5y + y2 = 0 + 9

Combine like terms: 0 + 9 = 9
4.5y + y2 = 9

The y term is 4.5y.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5y + 5.0625 + y2 = 9 + 5.0625

Reorder the terms:
5.0625 + 4.5y + y2 = 9 + 5.0625

Combine like terms: 9 + 5.0625 = 14.0625
5.0625 + 4.5y + y2 = 14.0625

Factor a perfect square on the left side:
(y + 2.25)(y + 2.25) = 14.0625

Calculate the square root of the right side: 3.75

Break this problem into two subproblems by setting 
(y + 2.25) equal to 3.75 and -3.75.

Subproblem 1
y + 2.25 = 3.75

Simplifying
y + 2.25 = 3.75

Reorder the terms:
2.25 + y = 3.75

Solving
2.25 + y = 3.75

Solving for variable 'y'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + y = 3.75 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + y = 3.75 + -2.25
y = 3.75 + -2.25

Combine like terms: 3.75 + -2.25 = 1.5
y = 1.5

Simplifying
y = 1.5

Subproblem 2
y + 2.25 = -3.75

Simplifying
y + 2.25 = -3.75

Reorder the terms:
2.25 + y = -3.75

Solving
2.25 + y = -3.75

Solving for variable 'y'.

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

Add '-2.25' to each side of the equation.
2.25 + -2.25 + y = -3.75 + -2.25

Combine like terms: 2.25 + -2.25 = 0.00
0.00 + y = -3.75 + -2.25
y = -3.75 + -2.25

Combine like terms: -3.75 + -2.25 = -6
y = -6

Simplifying
y = -6

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {1.5, -6}