3n^2+11n-500=0

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Solution for 3n^2+11n-500=0 equation:


Simplifying
3n2 + 11n + -500 = 0

Reorder the terms:
-500 + 11n + 3n2 = 0

Solving
-500 + 11n + 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'.
-166.6666667 + 3.666666667n + n2 = 0

Move the constant term to the right:

Add '166.6666667' to each side of the equation.
-166.6666667 + 3.666666667n + 166.6666667 + n2 = 0 + 166.6666667

Reorder the terms:
-166.6666667 + 166.6666667 + 3.666666667n + n2 = 0 + 166.6666667

Combine like terms: -166.6666667 + 166.6666667 = 0.0000000
0.0000000 + 3.666666667n + n2 = 0 + 166.6666667
3.666666667n + n2 = 0 + 166.6666667

Combine like terms: 0 + 166.6666667 = 166.6666667
3.666666667n + n2 = 166.6666667

The n term is 3.666666667n.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667n + 3.361111114 + n2 = 166.6666667 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667n + n2 = 166.6666667 + 3.361111114

Combine like terms: 166.6666667 + 3.361111114 = 170.027777814
3.361111114 + 3.666666667n + n2 = 170.027777814

Factor a perfect square on the left side:
(n + 1.833333334)(n + 1.833333334) = 170.027777814

Calculate the square root of the right side: 13.039469997

Break this problem into two subproblems by setting 
(n + 1.833333334) equal to 13.039469997 and -13.039469997.

Subproblem 1

n + 1.833333334 = 13.039469997 Simplifying n + 1.833333334 = 13.039469997 Reorder the terms: 1.833333334 + n = 13.039469997 Solving 1.833333334 + n = 13.039469997 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = 13.039469997 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = 13.039469997 + -1.833333334 n = 13.039469997 + -1.833333334 Combine like terms: 13.039469997 + -1.833333334 = 11.206136663 n = 11.206136663 Simplifying n = 11.206136663

Subproblem 2

n + 1.833333334 = -13.039469997 Simplifying n + 1.833333334 = -13.039469997 Reorder the terms: 1.833333334 + n = -13.039469997 Solving 1.833333334 + n = -13.039469997 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = -13.039469997 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = -13.039469997 + -1.833333334 n = -13.039469997 + -1.833333334 Combine like terms: -13.039469997 + -1.833333334 = -14.872803331 n = -14.872803331 Simplifying n = -14.872803331

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.206136663, -14.872803331}

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