y^2+9y=313.29

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Solution for y^2+9y=313.29 equation: 


Simplifying
y2 + 9y = 313.29

Reorder the terms:
9y + y2 = 313.29

Solving
9y + y2 = 313.29

Solving for variable 'y'.

Reorder the terms:
-313.29 + 9y + y2 = 313.29 + -313.29

Combine like terms: 313.29 + -313.29 = 0.00
-313.29 + 9y + y2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '313.29' to each side of the equation.
-313.29 + 9y + 313.29 + y2 = 0.00 + 313.29

Reorder the terms:
-313.29 + 313.29 + 9y + y2 = 0.00 + 313.29

Combine like terms: -313.29 + 313.29 = 0.00
0.00 + 9y + y2 = 0.00 + 313.29
9y + y2 = 0.00 + 313.29

Combine like terms: 0.00 + 313.29 = 313.29
9y + y2 = 313.29

The y term is 9y.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9y + 20.25 + y2 = 313.29 + 20.25

Reorder the terms:
20.25 + 9y + y2 = 313.29 + 20.25

Combine like terms: 313.29 + 20.25 = 333.54
20.25 + 9y + y2 = 333.54

Factor a perfect square on the left side:
(y + 4.5)(y + 4.5) = 333.54

Calculate the square root of the right side: 18.263077506

Break this problem into two subproblems by setting 
(y + 4.5) equal to 18.263077506 and -18.263077506.


Subproblem 1
y + 4.5 = 18.263077506

Simplifying
y + 4.5 = 18.263077506

Reorder the terms:
4.5 + y = 18.263077506

Solving
4.5 + y = 18.263077506

Solving for variable 'y'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + y = 18.263077506 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = 18.263077506 + -4.5
y = 18.263077506 + -4.5

Combine like terms: 18.263077506 + -4.5 = 13.763077506
y = 13.763077506

Simplifying
y = 13.763077506


Subproblem 2
y + 4.5 = -18.263077506

Simplifying
y + 4.5 = -18.263077506

Reorder the terms:
4.5 + y = -18.263077506

Solving
4.5 + y = -18.263077506

Solving for variable 'y'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + y = -18.263077506 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = -18.263077506 + -4.5
y = -18.263077506 + -4.5

Combine like terms: -18.263077506 + -4.5 = -22.763077506
y = -22.763077506

Simplifying
y = -22.763077506


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {13.763077506, -22.763077506}

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