3n^2-71n+24=0

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Solution for 3n^2-71n+24=0 equation:


Simplifying
3n2 + -71n + 24 = 0

Reorder the terms:
24 + -71n + 3n2 = 0

Solving
24 + -71n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
8 + -23.66666667n + n2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -23.66666667n + -8 + n2 = 0 + -8

Reorder the terms:
8 + -8 + -23.66666667n + n2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + -23.66666667n + n2 = 0 + -8
-23.66666667n + n2 = 0 + -8

Combine like terms: 0 + -8 = -8
-23.66666667n + n2 = -8

The n term is -23.66666667n.  Take half its coefficient (-11.83333334).
Square it (140.0277779) and add it to both sides.

Add '140.0277779' to each side of the equation.
-23.66666667n + 140.0277779 + n2 = -8 + 140.0277779

Reorder the terms:
140.0277779 + -23.66666667n + n2 = -8 + 140.0277779

Combine like terms: -8 + 140.0277779 = 132.0277779
140.0277779 + -23.66666667n + n2 = 132.0277779

Factor a perfect square on the left side:
(n + -11.83333334)(n + -11.83333334) = 132.0277779

Calculate the square root of the right side: 11.490334107

Break this problem into two subproblems by setting 
(n + -11.83333334) equal to 11.490334107 and -11.490334107.

Subproblem 1

n + -11.83333334 = 11.490334107 Simplifying n + -11.83333334 = 11.490334107 Reorder the terms: -11.83333334 + n = 11.490334107 Solving -11.83333334 + n = 11.490334107 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.83333334' to each side of the equation. -11.83333334 + 11.83333334 + n = 11.490334107 + 11.83333334 Combine like terms: -11.83333334 + 11.83333334 = 0.00000000 0.00000000 + n = 11.490334107 + 11.83333334 n = 11.490334107 + 11.83333334 Combine like terms: 11.490334107 + 11.83333334 = 23.323667447 n = 23.323667447 Simplifying n = 23.323667447

Subproblem 2

n + -11.83333334 = -11.490334107 Simplifying n + -11.83333334 = -11.490334107 Reorder the terms: -11.83333334 + n = -11.490334107 Solving -11.83333334 + n = -11.490334107 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.83333334' to each side of the equation. -11.83333334 + 11.83333334 + n = -11.490334107 + 11.83333334 Combine like terms: -11.83333334 + 11.83333334 = 0.00000000 0.00000000 + n = -11.490334107 + 11.83333334 n = -11.490334107 + 11.83333334 Combine like terms: -11.490334107 + 11.83333334 = 0.342999233 n = 0.342999233 Simplifying n = 0.342999233

Solution

The solution to the problem is based on the solutions from the subproblems. n = {23.323667447, 0.342999233}

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