y^2+9y=1

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Solution for y^2+9y=1 equation: 


Simplifying
y2 + 9y = 1

Reorder the terms:
9y + y2 = 1

Solving
9y + y2 = 1

Solving for variable 'y'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 9y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 9y + y2 = 0 + 1

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

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

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 = 1 + 20.25

Reorder the terms:
20.25 + 9y + y2 = 1 + 20.25

Combine like terms: 1 + 20.25 = 21.25
20.25 + 9y + y2 = 21.25

Factor a perfect square on the left side:
(y + 4.5)(y + 4.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(y + 4.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
y + 4.5 = 4.609772229

Simplifying
y + 4.5 = 4.609772229

Reorder the terms:
4.5 + y = 4.609772229

Solving
4.5 + y = 4.609772229

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 = 4.609772229 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = 4.609772229 + -4.5
y = 4.609772229 + -4.5

Combine like terms: 4.609772229 + -4.5 = 0.109772229
y = 0.109772229

Simplifying
y = 0.109772229


Subproblem 2
y + 4.5 = -4.609772229

Simplifying
y + 4.5 = -4.609772229

Reorder the terms:
4.5 + y = -4.609772229

Solving
4.5 + y = -4.609772229

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 = -4.609772229 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + y = -4.609772229 + -4.5
y = -4.609772229 + -4.5

Combine like terms: -4.609772229 + -4.5 = -9.109772229
y = -9.109772229

Simplifying
y = -9.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.109772229, -9.109772229}

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