9y^2+13=24y

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Solution for 9y^2+13=24y equation:


Simplifying
9y2 + 13 = 24y

Reorder the terms:
13 + 9y2 = 24y

Solving
13 + 9y2 = 24y

Solving for variable 'y'.

Reorder the terms:
13 + -24y + 9y2 = 24y + -24y

Combine like terms: 24y + -24y = 0
13 + -24y + 9y2 = 0

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

Divide each side by '9'.
1.444444444 + -2.666666667y + y2 = 0

Move the constant term to the right:

Add '-1.444444444' to each side of the equation.
1.444444444 + -2.666666667y + -1.444444444 + y2 = 0 + -1.444444444

Reorder the terms:
1.444444444 + -1.444444444 + -2.666666667y + y2 = 0 + -1.444444444

Combine like terms: 1.444444444 + -1.444444444 = 0.000000000
0.000000000 + -2.666666667y + y2 = 0 + -1.444444444
-2.666666667y + y2 = 0 + -1.444444444

Combine like terms: 0 + -1.444444444 = -1.444444444
-2.666666667y + y2 = -1.444444444

The y term is -2.666666667y.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667y + 1.777777780 + y2 = -1.444444444 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667y + y2 = -1.444444444 + 1.777777780

Combine like terms: -1.444444444 + 1.777777780 = 0.333333336
1.777777780 + -2.666666667y + y2 = 0.333333336

Factor a perfect square on the left side:
(y + -1.333333334)(y + -1.333333334) = 0.333333336

Calculate the square root of the right side: 0.577350271

Break this problem into two subproblems by setting 
(y + -1.333333334) equal to 0.577350271 and -0.577350271.

Subproblem 1

y + -1.333333334 = 0.577350271 Simplifying y + -1.333333334 = 0.577350271 Reorder the terms: -1.333333334 + y = 0.577350271 Solving -1.333333334 + y = 0.577350271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = 0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = 0.577350271 + 1.333333334 y = 0.577350271 + 1.333333334 Combine like terms: 0.577350271 + 1.333333334 = 1.910683605 y = 1.910683605 Simplifying y = 1.910683605

Subproblem 2

y + -1.333333334 = -0.577350271 Simplifying y + -1.333333334 = -0.577350271 Reorder the terms: -1.333333334 + y = -0.577350271 Solving -1.333333334 + y = -0.577350271 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + y = -0.577350271 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + y = -0.577350271 + 1.333333334 y = -0.577350271 + 1.333333334 Combine like terms: -0.577350271 + 1.333333334 = 0.755983063 y = 0.755983063 Simplifying y = 0.755983063

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.910683605, 0.755983063}

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