12n=n^2+31

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Solution for 12n=n^2+31 equation:


Simplifying
12n = n2 + 31

Reorder the terms:
12n = 31 + n2

Solving
12n = 31 + n2

Solving for variable 'n'.

Reorder the terms:
-31 + 12n + -1n2 = 31 + n2 + -31 + -1n2

Reorder the terms:
-31 + 12n + -1n2 = 31 + -31 + n2 + -1n2

Combine like terms: 31 + -31 = 0
-31 + 12n + -1n2 = 0 + n2 + -1n2
-31 + 12n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-31 + 12n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
31 + -12n + n2 = 0

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + -12n + -31 + n2 = 0 + -31

Reorder the terms:
31 + -31 + -12n + n2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + -12n + n2 = 0 + -31
-12n + n2 = 0 + -31

Combine like terms: 0 + -31 = -31
-12n + n2 = -31

The n term is -12n.  Take half its coefficient (-6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
-12n + 36 + n2 = -31 + 36

Reorder the terms:
36 + -12n + n2 = -31 + 36

Combine like terms: -31 + 36 = 5
36 + -12n + n2 = 5

Factor a perfect square on the left side:
(n + -6)(n + -6) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(n + -6) equal to 2.236067978 and -2.236067978.

Subproblem 1

n + -6 = 2.236067978 Simplifying n + -6 = 2.236067978 Reorder the terms: -6 + n = 2.236067978 Solving -6 + n = 2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = 2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + n = 2.236067978 + 6 n = 2.236067978 + 6 Combine like terms: 2.236067978 + 6 = 8.236067978 n = 8.236067978 Simplifying n = 8.236067978

Subproblem 2

n + -6 = -2.236067978 Simplifying n + -6 = -2.236067978 Reorder the terms: -6 + n = -2.236067978 Solving -6 + n = -2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + n = -2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + n = -2.236067978 + 6 n = -2.236067978 + 6 Combine like terms: -2.236067978 + 6 = 3.763932022 n = 3.763932022 Simplifying n = 3.763932022

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.236067978, 3.763932022}

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