9n^2-4n-16=0

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Solution for 9n^2-4n-16=0 equation:


Simplifying
9n2 + -4n + -16 = 0

Reorder the terms:
-16 + -4n + 9n2 = 0

Solving
-16 + -4n + 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'.
-1.777777778 + -0.4444444444n + n2 = 0

Move the constant term to the right:

Add '1.777777778' to each side of the equation.
-1.777777778 + -0.4444444444n + 1.777777778 + n2 = 0 + 1.777777778

Reorder the terms:
-1.777777778 + 1.777777778 + -0.4444444444n + n2 = 0 + 1.777777778

Combine like terms: -1.777777778 + 1.777777778 = 0.000000000
0.000000000 + -0.4444444444n + n2 = 0 + 1.777777778
-0.4444444444n + n2 = 0 + 1.777777778

Combine like terms: 0 + 1.777777778 = 1.777777778
-0.4444444444n + n2 = 1.777777778

The n term is -0.4444444444n.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444n + 0.04938271604 + n2 = 1.777777778 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444n + n2 = 1.777777778 + 0.04938271604

Combine like terms: 1.777777778 + 0.04938271604 = 1.82716049404
0.04938271604 + -0.4444444444n + n2 = 1.82716049404

Factor a perfect square on the left side:
(n + -0.2222222222)(n + -0.2222222222) = 1.82716049404

Calculate the square root of the right side: 1.351725007

Break this problem into two subproblems by setting 
(n + -0.2222222222) equal to 1.351725007 and -1.351725007.

Subproblem 1

n + -0.2222222222 = 1.351725007 Simplifying n + -0.2222222222 = 1.351725007 Reorder the terms: -0.2222222222 + n = 1.351725007 Solving -0.2222222222 + n = 1.351725007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = 1.351725007 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = 1.351725007 + 0.2222222222 n = 1.351725007 + 0.2222222222 Combine like terms: 1.351725007 + 0.2222222222 = 1.5739472292 n = 1.5739472292 Simplifying n = 1.5739472292

Subproblem 2

n + -0.2222222222 = -1.351725007 Simplifying n + -0.2222222222 = -1.351725007 Reorder the terms: -0.2222222222 + n = -1.351725007 Solving -0.2222222222 + n = -1.351725007 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2222222222' to each side of the equation. -0.2222222222 + 0.2222222222 + n = -1.351725007 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + n = -1.351725007 + 0.2222222222 n = -1.351725007 + 0.2222222222 Combine like terms: -1.351725007 + 0.2222222222 = -1.1295027848 n = -1.1295027848 Simplifying n = -1.1295027848

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.5739472292, -1.1295027848}

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