y^2+10y+20=0

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Solution for y^2+10y+20=0 equation: 


Simplifying
y2 + 10y + 20 = 0

Reorder the terms:
20 + 10y + y2 = 0

Solving
20 + 10y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + 10y + -20 + y2 = 0 + -20

Reorder the terms:
20 + -20 + 10y + y2 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + 10y + y2 = 0 + -20
10y + y2 = 0 + -20

Combine like terms: 0 + -20 = -20
10y + y2 = -20

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 = -20 + 25

Reorder the terms:
25 + 10y + y2 = -20 + 25

Combine like terms: -20 + 25 = 5
25 + 10y + y2 = 5

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

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(y + 5) equal to 2.236067978 and -2.236067978.


Subproblem 1
y + 5 = 2.236067978

Simplifying
y + 5 = 2.236067978

Reorder the terms:
5 + y = 2.236067978

Solving
5 + y = 2.236067978

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 = 2.236067978 + -5

Combine like terms: 5 + -5 = 0
0 + y = 2.236067978 + -5
y = 2.236067978 + -5

Combine like terms: 2.236067978 + -5 = -2.763932022
y = -2.763932022

Simplifying
y = -2.763932022


Subproblem 2
y + 5 = -2.236067978

Simplifying
y + 5 = -2.236067978

Reorder the terms:
5 + y = -2.236067978

Solving
5 + y = -2.236067978

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 = -2.236067978 + -5

Combine like terms: 5 + -5 = 0
0 + y = -2.236067978 + -5
y = -2.236067978 + -5

Combine like terms: -2.236067978 + -5 = -7.236067978
y = -7.236067978

Simplifying
y = -7.236067978


Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {-2.763932022, -7.236067978}

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