3y+y^2=90

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Solution for 3y+y^2=90 equation:


Simplifying
3y + y2 = 90

Solving
3y + y2 = 90

Solving for variable 'y'.

Reorder the terms:
-90 + 3y + y2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 3y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + 3y + 90 + y2 = 0 + 90

Reorder the terms:
-90 + 90 + 3y + y2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + 3y + y2 = 0 + 90
3y + y2 = 0 + 90

Combine like terms: 0 + 90 = 90
3y + y2 = 90

The y term is 3y.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3y + 2.25 + y2 = 90 + 2.25

Reorder the terms:
2.25 + 3y + y2 = 90 + 2.25

Combine like terms: 90 + 2.25 = 92.25
2.25 + 3y + y2 = 92.25

Factor a perfect square on the left side:
(y + 1.5)(y + 1.5) = 92.25

Calculate the square root of the right side: 9.604686356

Break this problem into two subproblems by setting 
(y + 1.5) equal to 9.604686356 and -9.604686356.

Subproblem 1

y + 1.5 = 9.604686356 Simplifying y + 1.5 = 9.604686356 Reorder the terms: 1.5 + y = 9.604686356 Solving 1.5 + y = 9.604686356 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 9.604686356 + -1.5 y = 9.604686356 + -1.5 Combine like terms: 9.604686356 + -1.5 = 8.104686356 y = 8.104686356 Simplifying y = 8.104686356

Subproblem 2

y + 1.5 = -9.604686356 Simplifying y + 1.5 = -9.604686356 Reorder the terms: 1.5 + y = -9.604686356 Solving 1.5 + y = -9.604686356 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -9.604686356 + -1.5 y = -9.604686356 + -1.5 Combine like terms: -9.604686356 + -1.5 = -11.104686356 y = -11.104686356 Simplifying y = -11.104686356

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.104686356, -11.104686356}

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