n^2-10n+22=2

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Solution for n^2-10n+22=2 equation:


Simplifying
n2 + -10n + 22 = 2

Reorder the terms:
22 + -10n + n2 = 2

Solving
22 + -10n + n2 = 2

Solving for variable 'n'.

Reorder the terms:
22 + -2 + -10n + n2 = 2 + -2

Combine like terms: 22 + -2 = 20
20 + -10n + n2 = 2 + -2

Combine like terms: 2 + -2 = 0
20 + -10n + n2 = 0

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 = 7.236067978 n = 7.236067978 Simplifying n = 7.236067978

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 = 2.763932022 n = 2.763932022 Simplifying n = 2.763932022

Solution

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

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