11n^2-5n=12

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Solution for 11n^2-5n=12 equation:


Simplifying
11n2 + -5n = 12

Reorder the terms:
-5n + 11n2 = 12

Solving
-5n + 11n2 = 12

Solving for variable 'n'.

Reorder the terms:
-12 + -5n + 11n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + -5n + 11n2 = 0

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

Divide each side by '11'.
-1.090909091 + -0.4545454545n + n2 = 0

Move the constant term to the right:

Add '1.090909091' to each side of the equation.
-1.090909091 + -0.4545454545n + 1.090909091 + n2 = 0 + 1.090909091

Reorder the terms:
-1.090909091 + 1.090909091 + -0.4545454545n + n2 = 0 + 1.090909091

Combine like terms: -1.090909091 + 1.090909091 = 0.000000000
0.000000000 + -0.4545454545n + n2 = 0 + 1.090909091
-0.4545454545n + n2 = 0 + 1.090909091

Combine like terms: 0 + 1.090909091 = 1.090909091
-0.4545454545n + n2 = 1.090909091

The n term is -0.4545454545n.  Take half its coefficient (-0.2272727273).
Square it (0.05165289257) and add it to both sides.

Add '0.05165289257' to each side of the equation.
-0.4545454545n + 0.05165289257 + n2 = 1.090909091 + 0.05165289257

Reorder the terms:
0.05165289257 + -0.4545454545n + n2 = 1.090909091 + 0.05165289257

Combine like terms: 1.090909091 + 0.05165289257 = 1.14256198357
0.05165289257 + -0.4545454545n + n2 = 1.14256198357

Factor a perfect square on the left side:
(n + -0.2272727273)(n + -0.2272727273) = 1.14256198357

Calculate the square root of the right side: 1.068906911

Break this problem into two subproblems by setting 
(n + -0.2272727273) equal to 1.068906911 and -1.068906911.

Subproblem 1

n + -0.2272727273 = 1.068906911 Simplifying n + -0.2272727273 = 1.068906911 Reorder the terms: -0.2272727273 + n = 1.068906911 Solving -0.2272727273 + n = 1.068906911 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2272727273' to each side of the equation. -0.2272727273 + 0.2272727273 + n = 1.068906911 + 0.2272727273 Combine like terms: -0.2272727273 + 0.2272727273 = 0.0000000000 0.0000000000 + n = 1.068906911 + 0.2272727273 n = 1.068906911 + 0.2272727273 Combine like terms: 1.068906911 + 0.2272727273 = 1.2961796383 n = 1.2961796383 Simplifying n = 1.2961796383

Subproblem 2

n + -0.2272727273 = -1.068906911 Simplifying n + -0.2272727273 = -1.068906911 Reorder the terms: -0.2272727273 + n = -1.068906911 Solving -0.2272727273 + n = -1.068906911 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2272727273' to each side of the equation. -0.2272727273 + 0.2272727273 + n = -1.068906911 + 0.2272727273 Combine like terms: -0.2272727273 + 0.2272727273 = 0.0000000000 0.0000000000 + n = -1.068906911 + 0.2272727273 n = -1.068906911 + 0.2272727273 Combine like terms: -1.068906911 + 0.2272727273 = -0.8416341837 n = -0.8416341837 Simplifying n = -0.8416341837

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.2961796383, -0.8416341837}

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