y^2+y=9

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Solution for y^2+y=9 equation:


Simplifying
y2 + y = 9

Reorder the terms:
y + y2 = 9

Solving
y + y2 = 9

Solving for variable 'y'.

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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.
y + 0.25 + y2 = 9 + 0.25

Reorder the terms:
0.25 + y + y2 = 9 + 0.25

Combine like terms: 9 + 0.25 = 9.25
0.25 + y + y2 = 9.25

Factor a perfect square on the left side:
(y + 0.5)(y + 0.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(y + 0.5) equal to 3.041381265 and -3.041381265.

Subproblem 1

y + 0.5 = 3.041381265 Simplifying y + 0.5 = 3.041381265 Reorder the terms: 0.5 + y = 3.041381265 Solving 0.5 + y = 3.041381265 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 = 3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 3.041381265 + -0.5 y = 3.041381265 + -0.5 Combine like terms: 3.041381265 + -0.5 = 2.541381265 y = 2.541381265 Simplifying y = 2.541381265

Subproblem 2

y + 0.5 = -3.041381265 Simplifying y + 0.5 = -3.041381265 Reorder the terms: 0.5 + y = -3.041381265 Solving 0.5 + y = -3.041381265 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 = -3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -3.041381265 + -0.5 y = -3.041381265 + -0.5 Combine like terms: -3.041381265 + -0.5 = -3.541381265 y = -3.541381265 Simplifying y = -3.541381265

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.541381265, -3.541381265}

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