n^2-6n+9=5

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Solution for n^2-6n+9=5 equation:


Simplifying
n2 + -6n + 9 = 5

Reorder the terms:
9 + -6n + n2 = 5

Solving
9 + -6n + n2 = 5

Solving for variable 'n'.

Reorder the terms:
9 + -5 + -6n + n2 = 5 + -5

Combine like terms: 9 + -5 = 4
4 + -6n + n2 = 5 + -5

Combine like terms: 5 + -5 = 0
4 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -6n + -4 + n2 = 0 + -4

Reorder the terms:
4 + -4 + -6n + n2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -6n + n2 = 0 + -4
-6n + n2 = 0 + -4

Combine like terms: 0 + -4 = -4
-6n + n2 = -4

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = -4 + 9

Reorder the terms:
9 + -6n + n2 = -4 + 9

Combine like terms: -4 + 9 = 5
9 + -6n + n2 = 5

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

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(n + -3) equal to 2.236067978 and -2.236067978.

Subproblem 1

n + -3 = 2.236067978 Simplifying n + -3 = 2.236067978 Reorder the terms: -3 + n = 2.236067978 Solving -3 + n = 2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.236067978 + 3 n = 2.236067978 + 3 Combine like terms: 2.236067978 + 3 = 5.236067978 n = 5.236067978 Simplifying n = 5.236067978

Subproblem 2

n + -3 = -2.236067978 Simplifying n + -3 = -2.236067978 Reorder the terms: -3 + n = -2.236067978 Solving -3 + n = -2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.236067978 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.236067978 + 3 n = -2.236067978 + 3 Combine like terms: -2.236067978 + 3 = 0.763932022 n = 0.763932022 Simplifying n = 0.763932022

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.236067978, 0.763932022}

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