9y^2-15y=y+9

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Solution for 9y^2-15y=y+9 equation:


Simplifying
9y2 + -15y = y + 9

Reorder the terms:
-15y + 9y2 = y + 9

Reorder the terms:
-15y + 9y2 = 9 + y

Solving
-15y + 9y2 = 9 + y

Solving for variable 'y'.

Reorder the terms:
-9 + -15y + -1y + 9y2 = 9 + y + -9 + -1y

Combine like terms: -15y + -1y = -16y
-9 + -16y + 9y2 = 9 + y + -9 + -1y

Reorder the terms:
-9 + -16y + 9y2 = 9 + -9 + y + -1y

Combine like terms: 9 + -9 = 0
-9 + -16y + 9y2 = 0 + y + -1y
-9 + -16y + 9y2 = y + -1y

Combine like terms: y + -1y = 0
-9 + -16y + 9y2 = 0

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

Divide each side by '9'.
-1 + -1.777777778y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1.777777778y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -1.777777778y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1.777777778y + y2 = 0 + 1
-1.777777778y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1.777777778y + y2 = 1

The y term is -1.777777778y.  Take half its coefficient (-0.888888889).
Square it (0.7901234570) and add it to both sides.

Add '0.7901234570' to each side of the equation.
-1.777777778y + 0.7901234570 + y2 = 1 + 0.7901234570

Reorder the terms:
0.7901234570 + -1.777777778y + y2 = 1 + 0.7901234570

Combine like terms: 1 + 0.7901234570 = 1.790123457
0.7901234570 + -1.777777778y + y2 = 1.790123457

Factor a perfect square on the left side:
(y + -0.888888889)(y + -0.888888889) = 1.790123457

Calculate the square root of the right side: 1.337954953

Break this problem into two subproblems by setting 
(y + -0.888888889) equal to 1.337954953 and -1.337954953.

Subproblem 1

y + -0.888888889 = 1.337954953 Simplifying y + -0.888888889 = 1.337954953 Reorder the terms: -0.888888889 + y = 1.337954953 Solving -0.888888889 + y = 1.337954953 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + y = 1.337954953 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + y = 1.337954953 + 0.888888889 y = 1.337954953 + 0.888888889 Combine like terms: 1.337954953 + 0.888888889 = 2.226843842 y = 2.226843842 Simplifying y = 2.226843842

Subproblem 2

y + -0.888888889 = -1.337954953 Simplifying y + -0.888888889 = -1.337954953 Reorder the terms: -0.888888889 + y = -1.337954953 Solving -0.888888889 + y = -1.337954953 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + y = -1.337954953 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + y = -1.337954953 + 0.888888889 y = -1.337954953 + 0.888888889 Combine like terms: -1.337954953 + 0.888888889 = -0.449066064 y = -0.449066064 Simplifying y = -0.449066064

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.226843842, -0.449066064}

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