n^2-22n+121=13

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Solution for n^2-22n+121=13 equation:


Simplifying
n2 + -22n + 121 = 13

Reorder the terms:
121 + -22n + n2 = 13

Solving
121 + -22n + n2 = 13

Solving for variable 'n'.

Reorder the terms:
121 + -13 + -22n + n2 = 13 + -13

Combine like terms: 121 + -13 = 108
108 + -22n + n2 = 13 + -13

Combine like terms: 13 + -13 = 0
108 + -22n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-108' to each side of the equation.
108 + -22n + -108 + n2 = 0 + -108

Reorder the terms:
108 + -108 + -22n + n2 = 0 + -108

Combine like terms: 108 + -108 = 0
0 + -22n + n2 = 0 + -108
-22n + n2 = 0 + -108

Combine like terms: 0 + -108 = -108
-22n + n2 = -108

The n term is -22n.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22n + 121 + n2 = -108 + 121

Reorder the terms:
121 + -22n + n2 = -108 + 121

Combine like terms: -108 + 121 = 13
121 + -22n + n2 = 13

Factor a perfect square on the left side:
(n + -11)(n + -11) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(n + -11) equal to 3.605551275 and -3.605551275.

Subproblem 1

n + -11 = 3.605551275 Simplifying n + -11 = 3.605551275 Reorder the terms: -11 + n = 3.605551275 Solving -11 + n = 3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = 3.605551275 + 11 Combine like terms: -11 + 11 = 0 0 + n = 3.605551275 + 11 n = 3.605551275 + 11 Combine like terms: 3.605551275 + 11 = 14.605551275 n = 14.605551275 Simplifying n = 14.605551275

Subproblem 2

n + -11 = -3.605551275 Simplifying n + -11 = -3.605551275 Reorder the terms: -11 + n = -3.605551275 Solving -11 + n = -3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = -3.605551275 + 11 Combine like terms: -11 + 11 = 0 0 + n = -3.605551275 + 11 n = -3.605551275 + 11 Combine like terms: -3.605551275 + 11 = 7.394448725 n = 7.394448725 Simplifying n = 7.394448725

Solution

The solution to the problem is based on the solutions from the subproblems. n = {14.605551275, 7.394448725}

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