9y^2-9y-52=0

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Solution for 9y^2-9y-52=0 equation: 


Simplifying
9y2 + -9y + -52 = 0

Reorder the terms:
-52 + -9y + 9y2 = 0

Solving
-52 + -9y + 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'.
-5.777777778 + -1y + y2 = 0

Move the constant term to the right:

Add '5.777777778' to each side of the equation.
-5.777777778 + -1y + 5.777777778 + y2 = 0 + 5.777777778

Reorder the terms:
-5.777777778 + 5.777777778 + -1y + y2 = 0 + 5.777777778

Combine like terms: -5.777777778 + 5.777777778 = 0.000000000
0.000000000 + -1y + y2 = 0 + 5.777777778
-1y + y2 = 0 + 5.777777778

Combine like terms: 0 + 5.777777778 = 5.777777778
-1y + y2 = 5.777777778

The y term is -1y.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1y + 0.25 + y2 = 5.777777778 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 5.777777778 + 0.25

Combine like terms: 5.777777778 + 0.25 = 6.027777778
0.25 + -1y + y2 = 6.027777778

Factor a perfect square on the left side:
(y + -0.5)(y + -0.5) = 6.027777778

Calculate the square root of the right side: 2.45515331

Break this problem into two subproblems by setting 
(y + -0.5) equal to 2.45515331 and -2.45515331.


Subproblem 1
y + -0.5 = 2.45515331

Simplifying
y + -0.5 = 2.45515331

Reorder the terms:
-0.5 + y = 2.45515331

Solving
-0.5 + y = 2.45515331

Solving for variable 'y'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + y = 2.45515331 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = 2.45515331 + 0.5
y = 2.45515331 + 0.5

Combine like terms: 2.45515331 + 0.5 = 2.95515331
y = 2.95515331

Simplifying
y = 2.95515331


Subproblem 2
y + -0.5 = -2.45515331

Simplifying
y + -0.5 = -2.45515331

Reorder the terms:
-0.5 + y = -2.45515331

Solving
-0.5 + y = -2.45515331

Solving for variable 'y'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + y = -2.45515331 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + y = -2.45515331 + 0.5
y = -2.45515331 + 0.5

Combine like terms: -2.45515331 + 0.5 = -1.95515331
y = -1.95515331

Simplifying
y = -1.95515331


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.95515331, -1.95515331}

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