12n^2-7n=108

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Solution for 12n^2-7n=108 equation:


Simplifying
12n2 + -7n = 108

Reorder the terms:
-7n + 12n2 = 108

Solving
-7n + 12n2 = 108

Solving for variable 'n'.

Reorder the terms:
-108 + -7n + 12n2 = 108 + -108

Combine like terms: 108 + -108 = 0
-108 + -7n + 12n2 = 0

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

Divide each side by '12'.
-9 + -0.5833333333n + n2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -0.5833333333n + 9 + n2 = 0 + 9

Reorder the terms:
-9 + 9 + -0.5833333333n + n2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -0.5833333333n + n2 = 0 + 9
-0.5833333333n + n2 = 0 + 9

Combine like terms: 0 + 9 = 9
-0.5833333333n + n2 = 9

The n term is -0.5833333333n.  Take half its coefficient (-0.2916666667).
Square it (0.08506944446) and add it to both sides.

Add '0.08506944446' to each side of the equation.
-0.5833333333n + 0.08506944446 + n2 = 9 + 0.08506944446

Reorder the terms:
0.08506944446 + -0.5833333333n + n2 = 9 + 0.08506944446

Combine like terms: 9 + 0.08506944446 = 9.08506944446
0.08506944446 + -0.5833333333n + n2 = 9.08506944446

Factor a perfect square on the left side:
(n + -0.2916666667)(n + -0.2916666667) = 9.08506944446

Calculate the square root of the right side: 3.014144894

Break this problem into two subproblems by setting 
(n + -0.2916666667) equal to 3.014144894 and -3.014144894.

Subproblem 1

n + -0.2916666667 = 3.014144894 Simplifying n + -0.2916666667 = 3.014144894 Reorder the terms: -0.2916666667 + n = 3.014144894 Solving -0.2916666667 + n = 3.014144894 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + n = 3.014144894 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + n = 3.014144894 + 0.2916666667 n = 3.014144894 + 0.2916666667 Combine like terms: 3.014144894 + 0.2916666667 = 3.3058115607 n = 3.3058115607 Simplifying n = 3.3058115607

Subproblem 2

n + -0.2916666667 = -3.014144894 Simplifying n + -0.2916666667 = -3.014144894 Reorder the terms: -0.2916666667 + n = -3.014144894 Solving -0.2916666667 + n = -3.014144894 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + n = -3.014144894 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + n = -3.014144894 + 0.2916666667 n = -3.014144894 + 0.2916666667 Combine like terms: -3.014144894 + 0.2916666667 = -2.7224782273 n = -2.7224782273 Simplifying n = -2.7224782273

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.3058115607, -2.7224782273}

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