9n^2+6n-37=0

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Solution for 9n^2+6n-37=0 equation: 


Simplifying
9n2 + 6n + -37 = 0

Reorder the terms:
-37 + 6n + 9n2 = 0

Solving
-37 + 6n + 9n2 = 0

Solving for variable 'n'.

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

Divide each side by '9'.
-4.111111111 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '4.111111111' to each side of the equation.
-4.111111111 + 0.6666666667n + 4.111111111 + n2 = 0 + 4.111111111

Reorder the terms:
-4.111111111 + 4.111111111 + 0.6666666667n + n2 = 0 + 4.111111111

Combine like terms: -4.111111111 + 4.111111111 = 0.000000000
0.000000000 + 0.6666666667n + n2 = 0 + 4.111111111
0.6666666667n + n2 = 0 + 4.111111111

Combine like terms: 0 + 4.111111111 = 4.111111111
0.6666666667n + n2 = 4.111111111

The n term is 0.6666666667n.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667n + 0.1111111112 + n2 = 4.111111111 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 4.111111111 + 0.1111111112

Combine like terms: 4.111111111 + 0.1111111112 = 4.2222222222
0.1111111112 + 0.6666666667n + n2 = 4.2222222222

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 4.2222222222

Calculate the square root of the right side: 2.054804668

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 2.054804668 and -2.054804668.


Subproblem 1
n + 0.3333333334 = 2.054804668

Simplifying
n + 0.3333333334 = 2.054804668

Reorder the terms:
0.3333333334 + n = 2.054804668

Solving
0.3333333334 + n = 2.054804668

Solving for variable 'n'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + n = 2.054804668 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + n = 2.054804668 + -0.3333333334
n = 2.054804668 + -0.3333333334

Combine like terms: 2.054804668 + -0.3333333334 = 1.7214713346
n = 1.7214713346

Simplifying
n = 1.7214713346


Subproblem 2
n + 0.3333333334 = -2.054804668

Simplifying
n + 0.3333333334 = -2.054804668

Reorder the terms:
0.3333333334 + n = -2.054804668

Solving
0.3333333334 + n = -2.054804668

Solving for variable 'n'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + n = -2.054804668 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + n = -2.054804668 + -0.3333333334
n = -2.054804668 + -0.3333333334

Combine like terms: -2.054804668 + -0.3333333334 = -2.3881380014
n = -2.3881380014

Simplifying
n = -2.3881380014


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.7214713346, -2.3881380014}

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