81y-9y^2=1

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Solution for 81y-9y^2=1 equation:


Simplifying
81y + -9y2 = 1

Solving
81y + -9y2 = 1

Solving for variable 'y'.

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

Combine like terms: 1 + -1 = 0
-1 + 81y + -9y2 = 0

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

Divide each side by '-9'.
0.1111111111 + -9y + y2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -9y + -0.1111111111 + y2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + -9y + y2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + -9y + y2 = 0 + -0.1111111111
-9y + y2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
-9y + y2 = -0.1111111111

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 = -0.1111111111 + 20.25

Reorder the terms:
20.25 + -9y + y2 = -0.1111111111 + 20.25

Combine like terms: -0.1111111111 + 20.25 = 20.1388888889
20.25 + -9y + y2 = 20.1388888889

Factor a perfect square on the left side:
(y + -4.5)(y + -4.5) = 20.1388888889

Calculate the square root of the right side: 4.487637339

Break this problem into two subproblems by setting 
(y + -4.5) equal to 4.487637339 and -4.487637339.

Subproblem 1

y + -4.5 = 4.487637339 Simplifying y + -4.5 = 4.487637339 Reorder the terms: -4.5 + y = 4.487637339 Solving -4.5 + y = 4.487637339 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 = 4.487637339 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = 4.487637339 + 4.5 y = 4.487637339 + 4.5 Combine like terms: 4.487637339 + 4.5 = 8.987637339 y = 8.987637339 Simplifying y = 8.987637339

Subproblem 2

y + -4.5 = -4.487637339 Simplifying y + -4.5 = -4.487637339 Reorder the terms: -4.5 + y = -4.487637339 Solving -4.5 + y = -4.487637339 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 = -4.487637339 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + y = -4.487637339 + 4.5 y = -4.487637339 + 4.5 Combine like terms: -4.487637339 + 4.5 = 0.012362661 y = 0.012362661 Simplifying y = 0.012362661

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.987637339, 0.012362661}

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