3n^2+7n-200=0

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Solution for 3n^2+7n-200=0 equation: 


Simplifying
3n2 + 7n + -200 = 0

Reorder the terms:
-200 + 7n + 3n2 = 0

Solving
-200 + 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'.
-66.66666667 + 2.333333333n + n2 = 0

Move the constant term to the right:

Add '66.66666667' to each side of the equation.
-66.66666667 + 2.333333333n + 66.66666667 + n2 = 0 + 66.66666667

Reorder the terms:
-66.66666667 + 66.66666667 + 2.333333333n + n2 = 0 + 66.66666667

Combine like terms: -66.66666667 + 66.66666667 = 0.00000000
0.00000000 + 2.333333333n + n2 = 0 + 66.66666667
2.333333333n + n2 = 0 + 66.66666667

Combine like terms: 0 + 66.66666667 = 66.66666667
2.333333333n + n2 = 66.66666667

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 = 66.66666667 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333n + n2 = 66.66666667 + 1.361111112

Combine like terms: 66.66666667 + 1.361111112 = 68.027777782
1.361111112 + 2.333333333n + n2 = 68.027777782

Factor a perfect square on the left side:
(n + 1.166666667)(n + 1.166666667) = 68.027777782

Calculate the square root of the right side: 8.247895355

Break this problem into two subproblems by setting 
(n + 1.166666667) equal to 8.247895355 and -8.247895355.


Subproblem 1
n + 1.166666667 = 8.247895355

Simplifying
n + 1.166666667 = 8.247895355

Reorder the terms:
1.166666667 + n = 8.247895355

Solving
1.166666667 + n = 8.247895355

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 = 8.247895355 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + n = 8.247895355 + -1.166666667
n = 8.247895355 + -1.166666667

Combine like terms: 8.247895355 + -1.166666667 = 7.081228688
n = 7.081228688

Simplifying
n = 7.081228688


Subproblem 2
n + 1.166666667 = -8.247895355

Simplifying
n + 1.166666667 = -8.247895355

Reorder the terms:
1.166666667 + n = -8.247895355

Solving
1.166666667 + n = -8.247895355

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 = -8.247895355 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + n = -8.247895355 + -1.166666667
n = -8.247895355 + -1.166666667

Combine like terms: -8.247895355 + -1.166666667 = -9.414562022
n = -9.414562022

Simplifying
n = -9.414562022


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.081228688, -9.414562022}

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