3n^2-51n-8=0

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Solution for 3n^2-51n-8=0 equation:


Simplifying
3n2 + -51n + -8 = 0

Reorder the terms:
-8 + -51n + 3n2 = 0

Solving
-8 + -51n + 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'.
-2.666666667 + -17n + n2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -17n + 2.666666667 + n2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -17n + n2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -17n + n2 = 0 + 2.666666667
-17n + n2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-17n + n2 = 2.666666667

The n term is -17n.  Take half its coefficient (-8.5).
Square it (72.25) and add it to both sides.

Add '72.25' to each side of the equation.
-17n + 72.25 + n2 = 2.666666667 + 72.25

Reorder the terms:
72.25 + -17n + n2 = 2.666666667 + 72.25

Combine like terms: 2.666666667 + 72.25 = 74.916666667
72.25 + -17n + n2 = 74.916666667

Factor a perfect square on the left side:
(n + -8.5)(n + -8.5) = 74.916666667

Calculate the square root of the right side: 8.655441448

Break this problem into two subproblems by setting 
(n + -8.5) equal to 8.655441448 and -8.655441448.

Subproblem 1

n + -8.5 = 8.655441448 Simplifying n + -8.5 = 8.655441448 Reorder the terms: -8.5 + n = 8.655441448 Solving -8.5 + n = 8.655441448 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + n = 8.655441448 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + n = 8.655441448 + 8.5 n = 8.655441448 + 8.5 Combine like terms: 8.655441448 + 8.5 = 17.155441448 n = 17.155441448 Simplifying n = 17.155441448

Subproblem 2

n + -8.5 = -8.655441448 Simplifying n + -8.5 = -8.655441448 Reorder the terms: -8.5 + n = -8.655441448 Solving -8.5 + n = -8.655441448 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '8.5' to each side of the equation. -8.5 + 8.5 + n = -8.655441448 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + n = -8.655441448 + 8.5 n = -8.655441448 + 8.5 Combine like terms: -8.655441448 + 8.5 = -0.155441448 n = -0.155441448 Simplifying n = -0.155441448

Solution

The solution to the problem is based on the solutions from the subproblems. n = {17.155441448, -0.155441448}

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