9y^2-3y-1=0

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Solution for 9y^2-3y-1=0 equation:


Simplifying
9y2 + -3y + -1 = 0

Reorder the terms:
-1 + -3y + 9y2 = 0

Solving
-1 + -3y + 9y2 = 0

Solving for variable 'y'.

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

Divide each side by '9'.
-0.1111111111 + -0.3333333333y + y2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.3333333333y + 0.1111111111 + y2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.3333333333y + y2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.3333333333y + y2 = 0 + 0.1111111111
-0.3333333333y + y2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.3333333333y + y2 = 0.1111111111

The y term is -0.3333333333y.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333y + 0.02777777779 + y2 = 0.1111111111 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333y + y2 = 0.1111111111 + 0.02777777779

Combine like terms: 0.1111111111 + 0.02777777779 = 0.13888888889
0.02777777779 + -0.3333333333y + y2 = 0.13888888889

Factor a perfect square on the left side:
(y + -0.1666666667)(y + -0.1666666667) = 0.13888888889

Calculate the square root of the right side: 0.372677996

Break this problem into two subproblems by setting 
(y + -0.1666666667) equal to 0.372677996 and -0.372677996.


Subproblem 1
y + -0.1666666667 = 0.372677996

Simplifying
y + -0.1666666667 = 0.372677996

Reorder the terms:
-0.1666666667 + y = 0.372677996

Solving
-0.1666666667 + y = 0.372677996

Solving for variable 'y'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + y = 0.372677996 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + y = 0.372677996 + 0.1666666667
y = 0.372677996 + 0.1666666667

Combine like terms: 0.372677996 + 0.1666666667 = 0.5393446627
y = 0.5393446627

Simplifying
y = 0.5393446627


Subproblem 2
y + -0.1666666667 = -0.372677996

Simplifying
y + -0.1666666667 = -0.372677996

Reorder the terms:
-0.1666666667 + y = -0.372677996

Solving
-0.1666666667 + y = -0.372677996

Solving for variable 'y'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + y = -0.372677996 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + y = -0.372677996 + 0.1666666667
y = -0.372677996 + 0.1666666667

Combine like terms: -0.372677996 + 0.1666666667 = -0.2060113293
y = -0.2060113293

Simplifying
y = -0.2060113293


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {0.5393446627, -0.2060113293}

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