9y^2+23=30y

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Solution for 9y^2+23=30y equation:


Simplifying
9y2 + 23 = 30y

Reorder the terms:
23 + 9y2 = 30y

Solving
23 + 9y2 = 30y

Solving for variable 'y'.

Reorder the terms:
23 + -30y + 9y2 = 30y + -30y

Combine like terms: 30y + -30y = 0
23 + -30y + 9y2 = 0

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

Divide each side by '9'.
2.555555556 + -3.333333333y + y2 = 0

Move the constant term to the right:

Add '-2.555555556' to each side of the equation.
2.555555556 + -3.333333333y + -2.555555556 + y2 = 0 + -2.555555556

Reorder the terms:
2.555555556 + -2.555555556 + -3.333333333y + y2 = 0 + -2.555555556

Combine like terms: 2.555555556 + -2.555555556 = 0.000000000
0.000000000 + -3.333333333y + y2 = 0 + -2.555555556
-3.333333333y + y2 = 0 + -2.555555556

Combine like terms: 0 + -2.555555556 = -2.555555556
-3.333333333y + y2 = -2.555555556

The y term is -3.333333333y.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333y + 2.777777779 + y2 = -2.555555556 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333y + y2 = -2.555555556 + 2.777777779

Combine like terms: -2.555555556 + 2.777777779 = 0.222222223
2.777777779 + -3.333333333y + y2 = 0.222222223

Factor a perfect square on the left side:
(y + -1.666666667)(y + -1.666666667) = 0.222222223

Calculate the square root of the right side: 0.471404522

Break this problem into two subproblems by setting 
(y + -1.666666667) equal to 0.471404522 and -0.471404522.

Subproblem 1

y + -1.666666667 = 0.471404522 Simplifying y + -1.666666667 = 0.471404522 Reorder the terms: -1.666666667 + y = 0.471404522 Solving -1.666666667 + y = 0.471404522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = 0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = 0.471404522 + 1.666666667 y = 0.471404522 + 1.666666667 Combine like terms: 0.471404522 + 1.666666667 = 2.138071189 y = 2.138071189 Simplifying y = 2.138071189

Subproblem 2

y + -1.666666667 = -0.471404522 Simplifying y + -1.666666667 = -0.471404522 Reorder the terms: -1.666666667 + y = -0.471404522 Solving -1.666666667 + y = -0.471404522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + y = -0.471404522 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + y = -0.471404522 + 1.666666667 y = -0.471404522 + 1.666666667 Combine like terms: -0.471404522 + 1.666666667 = 1.195262145 y = 1.195262145 Simplifying y = 1.195262145

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.138071189, 1.195262145}

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