3n^2+65n+24=0

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Solution for 3n^2+65n+24=0 equation:


Simplifying
3n2 + 65n + 24 = 0

Reorder the terms:
24 + 65n + 3n2 = 0

Solving
24 + 65n + 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 + 21.66666667n + n2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 21.66666667n + -8 + n2 = 0 + -8

Reorder the terms:
8 + -8 + 21.66666667n + n2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 21.66666667n + n2 = 0 + -8
21.66666667n + n2 = 0 + -8

Combine like terms: 0 + -8 = -8
21.66666667n + n2 = -8

The n term is 21.66666667n.  Take half its coefficient (10.83333334).
Square it (117.3611113) and add it to both sides.

Add '117.3611113' to each side of the equation.
21.66666667n + 117.3611113 + n2 = -8 + 117.3611113

Reorder the terms:
117.3611113 + 21.66666667n + n2 = -8 + 117.3611113

Combine like terms: -8 + 117.3611113 = 109.3611113
117.3611113 + 21.66666667n + n2 = 109.3611113

Factor a perfect square on the left side:
(n + 10.83333334)(n + 10.83333334) = 109.3611113

Calculate the square root of the right side: 10.457586304

Break this problem into two subproblems by setting 
(n + 10.83333334) equal to 10.457586304 and -10.457586304.

Subproblem 1

n + 10.83333334 = 10.457586304 Simplifying n + 10.83333334 = 10.457586304 Reorder the terms: 10.83333334 + n = 10.457586304 Solving 10.83333334 + n = 10.457586304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.83333334' to each side of the equation. 10.83333334 + -10.83333334 + n = 10.457586304 + -10.83333334 Combine like terms: 10.83333334 + -10.83333334 = 0.00000000 0.00000000 + n = 10.457586304 + -10.83333334 n = 10.457586304 + -10.83333334 Combine like terms: 10.457586304 + -10.83333334 = -0.375747036 n = -0.375747036 Simplifying n = -0.375747036

Subproblem 2

n + 10.83333334 = -10.457586304 Simplifying n + 10.83333334 = -10.457586304 Reorder the terms: 10.83333334 + n = -10.457586304 Solving 10.83333334 + n = -10.457586304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.83333334' to each side of the equation. 10.83333334 + -10.83333334 + n = -10.457586304 + -10.83333334 Combine like terms: 10.83333334 + -10.83333334 = 0.00000000 0.00000000 + n = -10.457586304 + -10.83333334 n = -10.457586304 + -10.83333334 Combine like terms: -10.457586304 + -10.83333334 = -21.290919644 n = -21.290919644 Simplifying n = -21.290919644

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.375747036, -21.290919644}

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