24y+9y^2=16

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Solution for 24y+9y^2=16 equation:


Simplifying
24y + 9y2 = 16

Solving
24y + 9y2 = 16

Solving for variable 'y'.

Reorder the terms:
-16 + 24y + 9y2 = 16 + -16

Combine like terms: 16 + -16 = 0
-16 + 24y + 9y2 = 0

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

Divide each side by '9'.
-1.777777778 + 2.666666667y + y2 = 0

Move the constant term to the right:

Add '1.777777778' to each side of the equation.
-1.777777778 + 2.666666667y + 1.777777778 + y2 = 0 + 1.777777778

Reorder the terms:
-1.777777778 + 1.777777778 + 2.666666667y + y2 = 0 + 1.777777778

Combine like terms: -1.777777778 + 1.777777778 = 0.000000000
0.000000000 + 2.666666667y + y2 = 0 + 1.777777778
2.666666667y + y2 = 0 + 1.777777778

Combine like terms: 0 + 1.777777778 = 1.777777778
2.666666667y + y2 = 1.777777778

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 = 1.777777778 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667y + y2 = 1.777777778 + 1.777777780

Combine like terms: 1.777777778 + 1.777777780 = 3.555555558
1.777777780 + 2.666666667y + y2 = 3.555555558

Factor a perfect square on the left side:
(y + 1.333333334)(y + 1.333333334) = 3.555555558

Calculate the square root of the right side: 1.885618084

Break this problem into two subproblems by setting 
(y + 1.333333334) equal to 1.885618084 and -1.885618084.

Subproblem 1

y + 1.333333334 = 1.885618084 Simplifying y + 1.333333334 = 1.885618084 Reorder the terms: 1.333333334 + y = 1.885618084 Solving 1.333333334 + y = 1.885618084 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 = 1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + y = 1.885618084 + -1.333333334 y = 1.885618084 + -1.333333334 Combine like terms: 1.885618084 + -1.333333334 = 0.55228475 y = 0.55228475 Simplifying y = 0.55228475

Subproblem 2

y + 1.333333334 = -1.885618084 Simplifying y + 1.333333334 = -1.885618084 Reorder the terms: 1.333333334 + y = -1.885618084 Solving 1.333333334 + y = -1.885618084 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 = -1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + y = -1.885618084 + -1.333333334 y = -1.885618084 + -1.333333334 Combine like terms: -1.885618084 + -1.333333334 = -3.218951418 y = -3.218951418 Simplifying y = -3.218951418

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.55228475, -3.218951418}

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