3n^2+108=37n

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Solution for 3n^2+108=37n equation:


Simplifying
3n2 + 108 = 37n

Reorder the terms:
108 + 3n2 = 37n

Solving
108 + 3n2 = 37n

Solving for variable 'n'.

Reorder the terms:
108 + -37n + 3n2 = 37n + -37n

Combine like terms: 37n + -37n = 0
108 + -37n + 3n2 = 0

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

Divide each side by '3'.
36 + -12.33333333n + n2 = 0

Move the constant term to the right:

Add '-36' to each side of the equation.
36 + -12.33333333n + -36 + n2 = 0 + -36

Reorder the terms:
36 + -36 + -12.33333333n + n2 = 0 + -36

Combine like terms: 36 + -36 = 0
0 + -12.33333333n + n2 = 0 + -36
-12.33333333n + n2 = 0 + -36

Combine like terms: 0 + -36 = -36
-12.33333333n + n2 = -36

The n term is -12.33333333n.  Take half its coefficient (-6.166666665).
Square it (38.02777776) and add it to both sides.

Add '38.02777776' to each side of the equation.
-12.33333333n + 38.02777776 + n2 = -36 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333n + n2 = -36 + 38.02777776

Combine like terms: -36 + 38.02777776 = 2.02777776
38.02777776 + -12.33333333n + n2 = 2.02777776

Factor a perfect square on the left side:
(n + -6.166666665)(n + -6.166666665) = 2.02777776

Calculate the square root of the right side: 1.424000618

Break this problem into two subproblems by setting 
(n + -6.166666665) equal to 1.424000618 and -1.424000618.

Subproblem 1

n + -6.166666665 = 1.424000618 Simplifying n + -6.166666665 = 1.424000618 Reorder the terms: -6.166666665 + n = 1.424000618 Solving -6.166666665 + n = 1.424000618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + n = 1.424000618 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + n = 1.424000618 + 6.166666665 n = 1.424000618 + 6.166666665 Combine like terms: 1.424000618 + 6.166666665 = 7.590667283 n = 7.590667283 Simplifying n = 7.590667283

Subproblem 2

n + -6.166666665 = -1.424000618 Simplifying n + -6.166666665 = -1.424000618 Reorder the terms: -6.166666665 + n = -1.424000618 Solving -6.166666665 + n = -1.424000618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + n = -1.424000618 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + n = -1.424000618 + 6.166666665 n = -1.424000618 + 6.166666665 Combine like terms: -1.424000618 + 6.166666665 = 4.742666047 n = 4.742666047 Simplifying n = 4.742666047

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.590667283, 4.742666047}

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