3n^2-7n-1000=0

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Solution for 3n^2-7n-1000=0 equation: 


Simplifying
3n2 + -7n + -1000 = 0

Reorder the terms:
-1000 + -7n + 3n2 = 0

Solving
-1000 + -7n + 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 + -2.333333333n + n2 = 0

Move the constant term to the right:

Add '333.3333333' to each side of the equation.
-333.3333333 + -2.333333333n + 333.3333333 + n2 = 0 + 333.3333333

Reorder the terms:
-333.3333333 + 333.3333333 + -2.333333333n + n2 = 0 + 333.3333333

Combine like terms: -333.3333333 + 333.3333333 = 0.0000000
0.0000000 + -2.333333333n + n2 = 0 + 333.3333333
-2.333333333n + n2 = 0 + 333.3333333

Combine like terms: 0 + 333.3333333 = 333.3333333
-2.333333333n + n2 = 333.3333333

The n term is -2.333333333n.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333n + 1.361111112 + n2 = 333.3333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333n + n2 = 333.3333333 + 1.361111112

Combine like terms: 333.3333333 + 1.361111112 = 334.694444412
1.361111112 + -2.333333333n + n2 = 334.694444412

Factor a perfect square on the left side:
(n + -1.166666667)(n + -1.166666667) = 334.694444412

Calculate the square root of the right side: 18.294656171

Break this problem into two subproblems by setting 
(n + -1.166666667) equal to 18.294656171 and -18.294656171.


Subproblem 1
n + -1.166666667 = 18.294656171

Simplifying
n + -1.166666667 = 18.294656171

Reorder the terms:
-1.166666667 + n = 18.294656171

Solving
-1.166666667 + n = 18.294656171

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + n = 18.294656171 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + n = 18.294656171 + 1.166666667
n = 18.294656171 + 1.166666667

Combine like terms: 18.294656171 + 1.166666667 = 19.461322838
n = 19.461322838

Simplifying
n = 19.461322838


Subproblem 2
n + -1.166666667 = -18.294656171

Simplifying
n + -1.166666667 = -18.294656171

Reorder the terms:
-1.166666667 + n = -18.294656171

Solving
-1.166666667 + n = -18.294656171

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '1.166666667' to each side of the equation.
-1.166666667 + 1.166666667 + n = -18.294656171 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + n = -18.294656171 + 1.166666667
n = -18.294656171 + 1.166666667

Combine like terms: -18.294656171 + 1.166666667 = -17.127989504
n = -17.127989504

Simplifying
n = -17.127989504


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {19.461322838, -17.127989504}

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