9n^2+39n=37

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Solution for 9n^2+39n=37 equation:


Simplifying
9n2 + 39n = 37

Reorder the terms:
39n + 9n2 = 37

Solving
39n + 9n2 = 37

Solving for variable 'n'.

Reorder the terms:
-37 + 39n + 9n2 = 37 + -37

Combine like terms: 37 + -37 = 0
-37 + 39n + 9n2 = 0

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

Divide each side by '9'.
-4.111111111 + 4.333333333n + n2 = 0

Move the constant term to the right:

Add '4.111111111' to each side of the equation.
-4.111111111 + 4.333333333n + 4.111111111 + n2 = 0 + 4.111111111

Reorder the terms:
-4.111111111 + 4.111111111 + 4.333333333n + n2 = 0 + 4.111111111

Combine like terms: -4.111111111 + 4.111111111 = 0.000000000
0.000000000 + 4.333333333n + n2 = 0 + 4.111111111
4.333333333n + n2 = 0 + 4.111111111

Combine like terms: 0 + 4.111111111 = 4.111111111
4.333333333n + n2 = 4.111111111

The n term is 4.333333333n.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333n + 4.694444446 + n2 = 4.111111111 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333n + n2 = 4.111111111 + 4.694444446

Combine like terms: 4.111111111 + 4.694444446 = 8.805555557
4.694444446 + 4.333333333n + n2 = 8.805555557

Factor a perfect square on the left side:
(n + 2.166666667)(n + 2.166666667) = 8.805555557

Calculate the square root of the right side: 2.967415636

Break this problem into two subproblems by setting 
(n + 2.166666667) equal to 2.967415636 and -2.967415636.

Subproblem 1

n + 2.166666667 = 2.967415636 Simplifying n + 2.166666667 = 2.967415636 Reorder the terms: 2.166666667 + n = 2.967415636 Solving 2.166666667 + n = 2.967415636 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = 2.967415636 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = 2.967415636 + -2.166666667 n = 2.967415636 + -2.166666667 Combine like terms: 2.967415636 + -2.166666667 = 0.800748969 n = 0.800748969 Simplifying n = 0.800748969

Subproblem 2

n + 2.166666667 = -2.967415636 Simplifying n + 2.166666667 = -2.967415636 Reorder the terms: 2.166666667 + n = -2.967415636 Solving 2.166666667 + n = -2.967415636 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + n = -2.967415636 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + n = -2.967415636 + -2.166666667 n = -2.967415636 + -2.166666667 Combine like terms: -2.967415636 + -2.166666667 = -5.134082303 n = -5.134082303 Simplifying n = -5.134082303

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.800748969, -5.134082303}

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