3n^2+n-1000=0

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Solution for 3n^2+n-1000=0 equation:


Simplifying
3n2 + n + -1000 = 0

Reorder the terms:
-1000 + n + 3n2 = 0

Solving
-1000 + 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'.
-333.3333333 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '333.3333333' to each side of the equation.
-333.3333333 + 0.3333333333n + 333.3333333 + n2 = 0 + 333.3333333

Reorder the terms:
-333.3333333 + 333.3333333 + 0.3333333333n + n2 = 0 + 333.3333333

Combine like terms: -333.3333333 + 333.3333333 = 0.0000000
0.0000000 + 0.3333333333n + n2 = 0 + 333.3333333
0.3333333333n + n2 = 0 + 333.3333333

Combine like terms: 0 + 333.3333333 = 333.3333333
0.3333333333n + n2 = 333.3333333

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 = 333.3333333 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 333.3333333 + 0.25

Combine like terms: 333.3333333 + 0.25 = 333.5833333
0.25 + 0.3333333333n + n2 = 333.5833333

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 333.5833333

Calculate the square root of the right side: 18.264263831

Break this problem into two subproblems by setting 
(n + 0.5) equal to 18.264263831 and -18.264263831.

Subproblem 1

n + 0.5 = 18.264263831 Simplifying n + 0.5 = 18.264263831 Reorder the terms: 0.5 + n = 18.264263831 Solving 0.5 + n = 18.264263831 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 = 18.264263831 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 18.264263831 + -0.5 n = 18.264263831 + -0.5 Combine like terms: 18.264263831 + -0.5 = 17.764263831 n = 17.764263831 Simplifying n = 17.764263831

Subproblem 2

n + 0.5 = -18.264263831 Simplifying n + 0.5 = -18.264263831 Reorder the terms: 0.5 + n = -18.264263831 Solving 0.5 + n = -18.264263831 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 = -18.264263831 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -18.264263831 + -0.5 n = -18.264263831 + -0.5 Combine like terms: -18.264263831 + -0.5 = -18.764263831 n = -18.764263831 Simplifying n = -18.764263831

Solution

The solution to the problem is based on the solutions from the subproblems. n = {17.764263831, -18.764263831}

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