9n^2=9n+16

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Solution for 9n^2=9n+16 equation:


Simplifying
9n2 = 9n + 16

Reorder the terms:
9n2 = 16 + 9n

Solving
9n2 = 16 + 9n

Solving for variable 'n'.

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

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

Combine like terms: 16 + -16 = 0
-16 + -9n + 9n2 = 0 + 9n + -9n
-16 + -9n + 9n2 = 9n + -9n

Combine like terms: 9n + -9n = 0
-16 + -9n + 9n2 = 0

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

Divide each side by '9'.
-1.777777778 + -1n + n2 = 0

Move the constant term to the right:

Add '1.777777778' to each side of the equation.
-1.777777778 + -1n + 1.777777778 + n2 = 0 + 1.777777778

Reorder the terms:
-1.777777778 + 1.777777778 + -1n + n2 = 0 + 1.777777778

Combine like terms: -1.777777778 + 1.777777778 = 0.000000000
0.000000000 + -1n + n2 = 0 + 1.777777778
-1n + n2 = 0 + 1.777777778

Combine like terms: 0 + 1.777777778 = 1.777777778
-1n + n2 = 1.777777778

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 1.777777778 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 1.777777778 + 0.25

Combine like terms: 1.777777778 + 0.25 = 2.027777778
0.25 + -1n + n2 = 2.027777778

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 2.027777778

Calculate the square root of the right side: 1.424000624

Break this problem into two subproblems by setting 
(n + -0.5) equal to 1.424000624 and -1.424000624.

Subproblem 1

n + -0.5 = 1.424000624 Simplifying n + -0.5 = 1.424000624 Reorder the terms: -0.5 + n = 1.424000624 Solving -0.5 + n = 1.424000624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.424000624 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.424000624 + 0.5 n = 1.424000624 + 0.5 Combine like terms: 1.424000624 + 0.5 = 1.924000624 n = 1.924000624 Simplifying n = 1.924000624

Subproblem 2

n + -0.5 = -1.424000624 Simplifying n + -0.5 = -1.424000624 Reorder the terms: -0.5 + n = -1.424000624 Solving -0.5 + n = -1.424000624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.424000624 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.424000624 + 0.5 n = -1.424000624 + 0.5 Combine like terms: -1.424000624 + 0.5 = -0.924000624 n = -0.924000624 Simplifying n = -0.924000624

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.924000624, -0.924000624}

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