n^2+10n+20=0

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


Simplifying
n2 + 10n + 20 = 0

Reorder the terms:
20 + 10n + n2 = 0

Solving
20 + 10n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + 10n + -20 + n2 = 0 + -20

Reorder the terms:
20 + -20 + 10n + n2 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + 10n + n2 = 0 + -20
10n + n2 = 0 + -20

Combine like terms: 0 + -20 = -20
10n + n2 = -20

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

Add '25' to each side of the equation.
10n + 25 + n2 = -20 + 25

Reorder the terms:
25 + 10n + n2 = -20 + 25

Combine like terms: -20 + 25 = 5
25 + 10n + n2 = 5

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

Calculate the square root of the right side: 2.236067978

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

Subproblem 1

n + 5 = 2.236067978 Simplifying n + 5 = 2.236067978 Reorder the terms: 5 + n = 2.236067978 Solving 5 + n = 2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 2.236067978 + -5 Combine like terms: 5 + -5 = 0 0 + n = 2.236067978 + -5 n = 2.236067978 + -5 Combine like terms: 2.236067978 + -5 = -2.763932022 n = -2.763932022 Simplifying n = -2.763932022

Subproblem 2

n + 5 = -2.236067978 Simplifying n + 5 = -2.236067978 Reorder the terms: 5 + n = -2.236067978 Solving 5 + n = -2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -2.236067978 + -5 Combine like terms: 5 + -5 = 0 0 + n = -2.236067978 + -5 n = -2.236067978 + -5 Combine like terms: -2.236067978 + -5 = -7.236067978 n = -7.236067978 Simplifying n = -7.236067978

Solution

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

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