y^2+5y+2.5=0

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Solution for y^2+5y+2.5=0 equation: 


Simplifying
y2 + 5y + 2.5 = 0

Reorder the terms:
2.5 + 5y + y2 = 0

Solving
2.5 + 5y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 5y + -2.5 + y2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 5y + y2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 5y + y2 = 0 + -2.5
5y + y2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
5y + y2 = -2.5

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 = -2.5 + 6.25

Reorder the terms:
6.25 + 5y + y2 = -2.5 + 6.25

Combine like terms: -2.5 + 6.25 = 3.75
6.25 + 5y + y2 = 3.75

Factor a perfect square on the left side:
(y + 2.5)(y + 2.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(y + 2.5) equal to 1.936491673 and -1.936491673.


Subproblem 1
y + 2.5 = 1.936491673

Simplifying
y + 2.5 = 1.936491673

Reorder the terms:
2.5 + y = 1.936491673

Solving
2.5 + y = 1.936491673

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 = 1.936491673 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = 1.936491673 + -2.5
y = 1.936491673 + -2.5

Combine like terms: 1.936491673 + -2.5 = -0.563508327
y = -0.563508327

Simplifying
y = -0.563508327


Subproblem 2
y + 2.5 = -1.936491673

Simplifying
y + 2.5 = -1.936491673

Reorder the terms:
2.5 + y = -1.936491673

Solving
2.5 + y = -1.936491673

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 = -1.936491673 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + y = -1.936491673 + -2.5
y = -1.936491673 + -2.5

Combine like terms: -1.936491673 + -2.5 = -4.436491673
y = -4.436491673

Simplifying
y = -4.436491673


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.563508327, -4.436491673}

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