5y+y^2=9

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Solution for 5y+y^2=9 equation:


Simplifying
5y + y2 = 9

Solving
5y + y2 = 9

Solving for variable 'y'.

Reorder the terms:
-9 + 5y + y2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 5y + 9 + y2 = 0 + 9

Reorder the terms:
-9 + 9 + 5y + y2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 5y + y2 = 0 + 9
5y + y2 = 0 + 9

Combine like terms: 0 + 9 = 9
5y + y2 = 9

The y term is 5y.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5y + 6.25 + y2 = 9 + 6.25

Reorder the terms:
6.25 + 5y + y2 = 9 + 6.25

Combine like terms: 9 + 6.25 = 15.25
6.25 + 5y + y2 = 15.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(y + 2.5) equal to 3.905124838 and -3.905124838.

Subproblem 1

y + 2.5 = 3.905124838 Simplifying y + 2.5 = 3.905124838 Reorder the terms: 2.5 + y = 3.905124838 Solving 2.5 + y = 3.905124838 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + y = 3.905124838 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + y = 3.905124838 + -2.5 y = 3.905124838 + -2.5 Combine like terms: 3.905124838 + -2.5 = 1.405124838 y = 1.405124838 Simplifying y = 1.405124838

Subproblem 2

y + 2.5 = -3.905124838 Simplifying y + 2.5 = -3.905124838 Reorder the terms: 2.5 + y = -3.905124838 Solving 2.5 + y = -3.905124838 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + y = -3.905124838 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + y = -3.905124838 + -2.5 y = -3.905124838 + -2.5 Combine like terms: -3.905124838 + -2.5 = -6.405124838 y = -6.405124838 Simplifying y = -6.405124838

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.405124838, -6.405124838}

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