(72-y^2)+y=9

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Solution for (72-y^2)+y=9 equation:


Simplifying
(72 + -1y2) + y = 9

Remove parenthesis around (72 + -1y2)
72 + -1y2 + y = 9

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

Solving
72 + y + -1y2 = 9

Solving for variable 'y'.

Reorder the terms:
72 + -9 + y + -1y2 = 9 + -9

Combine like terms: 72 + -9 = 63
63 + y + -1y2 = 9 + -9

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-63 + -1y + y2 = 0

Move the constant term to the right:

Add '63' to each side of the equation.
-63 + -1y + 63 + y2 = 0 + 63

Reorder the terms:
-63 + 63 + -1y + y2 = 0 + 63

Combine like terms: -63 + 63 = 0
0 + -1y + y2 = 0 + 63
-1y + y2 = 0 + 63

Combine like terms: 0 + 63 = 63
-1y + y2 = 63

The y term is y.  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 = 63 + 0.25

Reorder the terms:
0.25 + -1y + y2 = 63 + 0.25

Combine like terms: 63 + 0.25 = 63.25
0.25 + -1y + y2 = 63.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 63.25

Calculate the square root of the right side: 7.95298686

Break this problem into two subproblems by setting 
(y + 0.5) equal to 7.95298686 and -7.95298686.

Subproblem 1

y + 0.5 = 7.95298686 Simplifying y + 0.5 = 7.95298686 Reorder the terms: 0.5 + y = 7.95298686 Solving 0.5 + y = 7.95298686 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 = 7.95298686 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 7.95298686 + -0.5 y = 7.95298686 + -0.5 Combine like terms: 7.95298686 + -0.5 = 7.45298686 y = 7.45298686 Simplifying y = 7.45298686

Subproblem 2

y + 0.5 = -7.95298686 Simplifying y + 0.5 = -7.95298686 Reorder the terms: 0.5 + y = -7.95298686 Solving 0.5 + y = -7.95298686 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 = -7.95298686 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -7.95298686 + -0.5 y = -7.95298686 + -0.5 Combine like terms: -7.95298686 + -0.5 = -8.45298686 y = -8.45298686 Simplifying y = -8.45298686

Solution

The solution to the problem is based on the solutions from the subproblems. y = {7.45298686, -8.45298686}

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