y^2+7y+5=0

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Solution for y^2+7y+5=0 equation:


Simplifying
y2 + 7y + 5 = 0

Reorder the terms:
5 + 7y + y2 = 0

Solving
5 + 7y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 7y + -5 + y2 = 0 + -5

Reorder the terms:
5 + -5 + 7y + y2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 7y + y2 = 0 + -5
7y + y2 = 0 + -5

Combine like terms: 0 + -5 = -5
7y + y2 = -5

The y term is 7y.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7y + 12.25 + y2 = -5 + 12.25

Reorder the terms:
12.25 + 7y + y2 = -5 + 12.25

Combine like terms: -5 + 12.25 = 7.25
12.25 + 7y + y2 = 7.25

Factor a perfect square on the left side:
(y + 3.5)(y + 3.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(y + 3.5) equal to 2.692582404 and -2.692582404.


Subproblem 1
y + 3.5 = 2.692582404

Simplifying
y + 3.5 = 2.692582404

Reorder the terms:
3.5 + y = 2.692582404

Solving
3.5 + y = 2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + y = 2.692582404 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = 2.692582404 + -3.5
y = 2.692582404 + -3.5

Combine like terms: 2.692582404 + -3.5 = -0.807417596
y = -0.807417596

Simplifying
y = -0.807417596


Subproblem 2
y + 3.5 = -2.692582404

Simplifying
y + 3.5 = -2.692582404

Reorder the terms:
3.5 + y = -2.692582404

Solving
3.5 + y = -2.692582404

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '-3.5' to each side of the equation.
3.5 + -3.5 + y = -2.692582404 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + y = -2.692582404 + -3.5
y = -2.692582404 + -3.5

Combine like terms: -2.692582404 + -3.5 = -6.192582404
y = -6.192582404

Simplifying
y = -6.192582404


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.807417596, -6.192582404}

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