n^2-10n+22=0

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Solution for n^2-10n+22=0 equation:


Simplifying
n2 + -10n + 22 = 0

Reorder the terms:
22 + -10n + n2 = 0

Solving
22 + -10n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-22' to each side of the equation.
22 + -10n + -22 + n2 = 0 + -22

Reorder the terms:
22 + -22 + -10n + n2 = 0 + -22

Combine like terms: 22 + -22 = 0
0 + -10n + n2 = 0 + -22
-10n + n2 = 0 + -22

Combine like terms: 0 + -22 = -22
-10n + n2 = -22

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 = -22 + 25

Reorder the terms:
25 + -10n + n2 = -22 + 25

Combine like terms: -22 + 25 = 3
25 + -10n + n2 = 3

Factor a perfect square on the left side:
(n + -5)(n + -5) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + -5) equal to 1.732050808 and -1.732050808.

Subproblem 1

n + -5 = 1.732050808 Simplifying n + -5 = 1.732050808 Reorder the terms: -5 + n = 1.732050808 Solving -5 + n = 1.732050808 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 = 1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + n = 1.732050808 + 5 n = 1.732050808 + 5 Combine like terms: 1.732050808 + 5 = 6.732050808 n = 6.732050808 Simplifying n = 6.732050808

Subproblem 2

n + -5 = -1.732050808 Simplifying n + -5 = -1.732050808 Reorder the terms: -5 + n = -1.732050808 Solving -5 + n = -1.732050808 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 = -1.732050808 + 5 Combine like terms: -5 + 5 = 0 0 + n = -1.732050808 + 5 n = -1.732050808 + 5 Combine like terms: -1.732050808 + 5 = 3.267949192 n = 3.267949192 Simplifying n = 3.267949192

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.732050808, 3.267949192}

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