n^2-18n+31=0

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Solution for n^2-18n+31=0 equation:


Simplifying
n2 + -18n + 31 = 0

Reorder the terms:
31 + -18n + n2 = 0

Solving
31 + -18n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + -18n + -31 + n2 = 0 + -31

Reorder the terms:
31 + -31 + -18n + n2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + -18n + n2 = 0 + -31
-18n + n2 = 0 + -31

Combine like terms: 0 + -31 = -31
-18n + n2 = -31

The n term is -18n.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18n + 81 + n2 = -31 + 81

Reorder the terms:
81 + -18n + n2 = -31 + 81

Combine like terms: -31 + 81 = 50
81 + -18n + n2 = 50

Factor a perfect square on the left side:
(n + -9)(n + -9) = 50

Calculate the square root of the right side: 7.071067812

Break this problem into two subproblems by setting 
(n + -9) equal to 7.071067812 and -7.071067812.

Subproblem 1

n + -9 = 7.071067812 Simplifying n + -9 = 7.071067812 Reorder the terms: -9 + n = 7.071067812 Solving -9 + n = 7.071067812 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 7.071067812 + 9 Combine like terms: -9 + 9 = 0 0 + n = 7.071067812 + 9 n = 7.071067812 + 9 Combine like terms: 7.071067812 + 9 = 16.071067812 n = 16.071067812 Simplifying n = 16.071067812

Subproblem 2

n + -9 = -7.071067812 Simplifying n + -9 = -7.071067812 Reorder the terms: -9 + n = -7.071067812 Solving -9 + n = -7.071067812 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -7.071067812 + 9 Combine like terms: -9 + 9 = 0 0 + n = -7.071067812 + 9 n = -7.071067812 + 9 Combine like terms: -7.071067812 + 9 = 1.928932188 n = 1.928932188 Simplifying n = 1.928932188

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.071067812, 1.928932188}

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