3n^2+n-318=0

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Solution for 3n^2+n-318=0 equation:


Simplifying
3n2 + n + -318 = 0

Reorder the terms:
-318 + n + 3n2 = 0

Solving
-318 + n + 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'.
-106 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '106' to each side of the equation.
-106 + 0.3333333333n + 106 + n2 = 0 + 106

Reorder the terms:
-106 + 106 + 0.3333333333n + n2 = 0 + 106

Combine like terms: -106 + 106 = 0
0 + 0.3333333333n + n2 = 0 + 106
0.3333333333n + n2 = 0 + 106

Combine like terms: 0 + 106 = 106
0.3333333333n + n2 = 106

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333n + 0.25 + n2 = 106 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 106 + 0.25

Combine like terms: 106 + 0.25 = 106.25
0.25 + 0.3333333333n + n2 = 106.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 106.25

Calculate the square root of the right side: 10.307764064

Break this problem into two subproblems by setting 
(n + 0.5) equal to 10.307764064 and -10.307764064.

Subproblem 1

n + 0.5 = 10.307764064 Simplifying n + 0.5 = 10.307764064 Reorder the terms: 0.5 + n = 10.307764064 Solving 0.5 + n = 10.307764064 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 10.307764064 + -0.5 n = 10.307764064 + -0.5 Combine like terms: 10.307764064 + -0.5 = 9.807764064 n = 9.807764064 Simplifying n = 9.807764064

Subproblem 2

n + 0.5 = -10.307764064 Simplifying n + 0.5 = -10.307764064 Reorder the terms: 0.5 + n = -10.307764064 Solving 0.5 + n = -10.307764064 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -10.307764064 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -10.307764064 + -0.5 n = -10.307764064 + -0.5 Combine like terms: -10.307764064 + -0.5 = -10.807764064 n = -10.807764064 Simplifying n = -10.807764064

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.807764064, -10.807764064}

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