y^2+5y=-2

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


Simplifying
y2 + 5y = -2

Reorder the terms:
5y + y2 = -2

Solving
5y + y2 = -2

Solving for variable 'y'.

Reorder the terms:
2 + 5y + y2 = -2 + 2

Combine like terms: -2 + 2 = 0
2 + 5y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5y + -2 + y2 = 0 + -2

Reorder the terms:
2 + -2 + 5y + y2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5y + y2 = 0 + -2
5y + y2 = 0 + -2

Combine like terms: 0 + -2 = -2
5y + y2 = -2

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 = -2 + 6.25

Reorder the terms:
6.25 + 5y + y2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + 5y + y2 = 4.25

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(y + 2.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
y + 2.5 = 2.061552813

Simplifying
y + 2.5 = 2.061552813

Reorder the terms:
2.5 + y = 2.061552813

Solving
2.5 + y = 2.061552813

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 = 2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 2.061552813 + -2.5
y = 2.061552813 + -2.5

Combine like terms: 2.061552813 + -2.5 = -0.438447187
y = -0.438447187

Simplifying
y = -0.438447187


Subproblem 2
y + 2.5 = -2.061552813

Simplifying
y + 2.5 = -2.061552813

Reorder the terms:
2.5 + y = -2.061552813

Solving
2.5 + y = -2.061552813

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 = -2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -2.061552813 + -2.5
y = -2.061552813 + -2.5

Combine like terms: -2.061552813 + -2.5 = -4.561552813
y = -4.561552813

Simplifying
y = -4.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.438447187, -4.561552813}

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