y^2+10y+7=0

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Solution for y^2+10y+7=0 equation: 


Simplifying
y2 + 10y + 7 = 0

Reorder the terms:
7 + 10y + y2 = 0

Solving
7 + 10y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 10y + -7 + y2 = 0 + -7

Reorder the terms:
7 + -7 + 10y + y2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 10y + y2 = 0 + -7
10y + y2 = 0 + -7

Combine like terms: 0 + -7 = -7
10y + y2 = -7

The y term is 10y.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10y + 25 + y2 = -7 + 25

Reorder the terms:
25 + 10y + y2 = -7 + 25

Combine like terms: -7 + 25 = 18
25 + 10y + y2 = 18

Factor a perfect square on the left side:
(y + 5)(y + 5) = 18

Calculate the square root of the right side: 4.242640687

Break this problem into two subproblems by setting 
(y + 5) equal to 4.242640687 and -4.242640687.


Subproblem 1
y + 5 = 4.242640687

Simplifying
y + 5 = 4.242640687

Reorder the terms:
5 + y = 4.242640687

Solving
5 + y = 4.242640687

Solving for variable 'y'.

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

Add '-5' to each side of the equation.
5 + -5 + y = 4.242640687 + -5

Combine like terms: 5 + -5 = 0
0 + y = 4.242640687 + -5
y = 4.242640687 + -5

Combine like terms: 4.242640687 + -5 = -0.757359313
y = -0.757359313

Simplifying
y = -0.757359313


Subproblem 2
y + 5 = -4.242640687

Simplifying
y + 5 = -4.242640687

Reorder the terms:
5 + y = -4.242640687

Solving
5 + y = -4.242640687

Solving for variable 'y'.

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

Add '-5' to each side of the equation.
5 + -5 + y = -4.242640687 + -5

Combine like terms: 5 + -5 = 0
0 + y = -4.242640687 + -5
y = -4.242640687 + -5

Combine like terms: -4.242640687 + -5 = -9.242640687
y = -9.242640687

Simplifying
y = -9.242640687


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-0.757359313, -9.242640687}

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