16n^2+36n-1000=0

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Solution for 16n^2+36n-1000=0 equation: 


Simplifying
16n2 + 36n + -1000 = 0

Reorder the terms:
-1000 + 36n + 16n2 = 0

Solving
-1000 + 36n + 16n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-250 + 9n + 4n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-250 + 9n + 4n2)' equal to zero and attempt to solve:

Simplifying
-250 + 9n + 4n2 = 0

Solving
-250 + 9n + 4n2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-62.5 + 2.25n + n2 = 0

Move the constant term to the right:

Add '62.5' to each side of the equation.
-62.5 + 2.25n + 62.5 + n2 = 0 + 62.5

Reorder the terms:
-62.5 + 62.5 + 2.25n + n2 = 0 + 62.5

Combine like terms: -62.5 + 62.5 = 0.0
0.0 + 2.25n + n2 = 0 + 62.5
2.25n + n2 = 0 + 62.5

Combine like terms: 0 + 62.5 = 62.5
2.25n + n2 = 62.5

The n term is 2.25n.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25n + 1.265625 + n2 = 62.5 + 1.265625

Reorder the terms:
1.265625 + 2.25n + n2 = 62.5 + 1.265625

Combine like terms: 62.5 + 1.265625 = 63.765625
1.265625 + 2.25n + n2 = 63.765625

Factor a perfect square on the left side:
(n + 1.125)(n + 1.125) = 63.765625

Calculate the square root of the right side: 7.985338127

Break this problem into two subproblems by setting 
(n + 1.125) equal to 7.985338127 and -7.985338127.


Subproblem 1
n + 1.125 = 7.985338127

Simplifying
n + 1.125 = 7.985338127

Reorder the terms:
1.125 + n = 7.985338127

Solving
1.125 + n = 7.985338127

Solving for variable 'n'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + n = 7.985338127 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + n = 7.985338127 + -1.125
n = 7.985338127 + -1.125

Combine like terms: 7.985338127 + -1.125 = 6.860338127
n = 6.860338127

Simplifying
n = 6.860338127


Subproblem 2
n + 1.125 = -7.985338127

Simplifying
n + 1.125 = -7.985338127

Reorder the terms:
1.125 + n = -7.985338127

Solving
1.125 + n = -7.985338127

Solving for variable 'n'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + n = -7.985338127 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + n = -7.985338127 + -1.125
n = -7.985338127 + -1.125

Combine like terms: -7.985338127 + -1.125 = -9.110338127
n = -9.110338127

Simplifying
n = -9.110338127


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.860338127, -9.110338127}
Solution
n = {6.860338127, -9.110338127}

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