9(1+y)(9-y)=1

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Solution for 9(1+y)(9-y)=1 equation: 


Simplifying
9(1 + y)(9 + -1y) = 1

Multiply (1 + y) * (9 + -1y)
9(1(9 + -1y) + y(9 + -1y)) = 1
9((9 * 1 + -1y * 1) + y(9 + -1y)) = 1
9((9 + -1y) + y(9 + -1y)) = 1
9(9 + -1y + (9 * y + -1y * y)) = 1
9(9 + -1y + (9y + -1y2)) = 1

Combine like terms: -1y + 9y = 8y
9(9 + 8y + -1y2) = 1
(9 * 9 + 8y * 9 + -1y2 * 9) = 1
(81 + 72y + -9y2) = 1

Solving
81 + 72y + -9y2 = 1

Solving for variable 'y'.

Reorder the terms:
81 + -1 + 72y + -9y2 = 1 + -1

Combine like terms: 81 + -1 = 80
80 + 72y + -9y2 = 1 + -1

Combine like terms: 1 + -1 = 0
80 + 72y + -9y2 = 0

Begin completing the square.  Divide all terms by
-9 the coefficient of the squared term: 

Divide each side by '-9'.
-8.888888889 + -8y + y2 = 0

Move the constant term to the right:

Add '8.888888889' to each side of the equation.
-8.888888889 + -8y + 8.888888889 + y2 = 0 + 8.888888889

Reorder the terms:
-8.888888889 + 8.888888889 + -8y + y2 = 0 + 8.888888889

Combine like terms: -8.888888889 + 8.888888889 = 0.000000000
0.000000000 + -8y + y2 = 0 + 8.888888889
-8y + y2 = 0 + 8.888888889

Combine like terms: 0 + 8.888888889 = 8.888888889
-8y + y2 = 8.888888889

The y term is -8y.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8y + 16 + y2 = 8.888888889 + 16

Reorder the terms:
16 + -8y + y2 = 8.888888889 + 16

Combine like terms: 8.888888889 + 16 = 24.888888889
16 + -8y + y2 = 24.888888889

Factor a perfect square on the left side:
(y + -4)(y + -4) = 24.888888889

Calculate the square root of the right side: 4.988876516

Break this problem into two subproblems by setting 
(y + -4) equal to 4.988876516 and -4.988876516.


Subproblem 1
y + -4 = 4.988876516

Simplifying
y + -4 = 4.988876516

Reorder the terms:
-4 + y = 4.988876516

Solving
-4 + y = 4.988876516

Solving for variable 'y'.

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

Add '4' to each side of the equation.
-4 + 4 + y = 4.988876516 + 4

Combine like terms: -4 + 4 = 0
0 + y = 4.988876516 + 4
y = 4.988876516 + 4

Combine like terms: 4.988876516 + 4 = 8.988876516
y = 8.988876516

Simplifying
y = 8.988876516


Subproblem 2
y + -4 = -4.988876516

Simplifying
y + -4 = -4.988876516

Reorder the terms:
-4 + y = -4.988876516

Solving
-4 + y = -4.988876516

Solving for variable 'y'.

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

Add '4' to each side of the equation.
-4 + 4 + y = -4.988876516 + 4

Combine like terms: -4 + 4 = 0
0 + y = -4.988876516 + 4
y = -4.988876516 + 4

Combine like terms: -4.988876516 + 4 = -0.988876516
y = -0.988876516

Simplifying
y = -0.988876516


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {8.988876516, -0.988876516}

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