8n(6n+5)=9

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Solution for 8n(6n+5)=9 equation:


Simplifying
8n(6n + 5) = 9

Reorder the terms:
8n(5 + 6n) = 9
(5 * 8n + 6n * 8n) = 9
(40n + 48n2) = 9

Solving
40n + 48n2 = 9

Solving for variable 'n'.

Reorder the terms:
-9 + 40n + 48n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 40n + 48n2 = 0

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

Divide each side by '48'.
-0.1875 + 0.8333333333n + n2 = 0

Move the constant term to the right:

Add '0.1875' to each side of the equation.
-0.1875 + 0.8333333333n + 0.1875 + n2 = 0 + 0.1875

Reorder the terms:
-0.1875 + 0.1875 + 0.8333333333n + n2 = 0 + 0.1875

Combine like terms: -0.1875 + 0.1875 = 0.0000
0.0000 + 0.8333333333n + n2 = 0 + 0.1875
0.8333333333n + n2 = 0 + 0.1875

Combine like terms: 0 + 0.1875 = 0.1875
0.8333333333n + n2 = 0.1875

The n term is 0.8333333333n.  Take half its coefficient (0.4166666667).
Square it (0.1736111111) and add it to both sides.

Add '0.1736111111' to each side of the equation.
0.8333333333n + 0.1736111111 + n2 = 0.1875 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333n + n2 = 0.1875 + 0.1736111111

Combine like terms: 0.1875 + 0.1736111111 = 0.3611111111
0.1736111111 + 0.8333333333n + n2 = 0.3611111111

Factor a perfect square on the left side:
(n + 0.4166666667)(n + 0.4166666667) = 0.3611111111

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(n + 0.4166666667) equal to 0.600925213 and -0.600925213.

Subproblem 1

n + 0.4166666667 = 0.600925213 Simplifying n + 0.4166666667 = 0.600925213 Reorder the terms: 0.4166666667 + n = 0.600925213 Solving 0.4166666667 + n = 0.600925213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = 0.600925213 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = 0.600925213 + -0.4166666667 n = 0.600925213 + -0.4166666667 Combine like terms: 0.600925213 + -0.4166666667 = 0.1842585463 n = 0.1842585463 Simplifying n = 0.1842585463

Subproblem 2

n + 0.4166666667 = -0.600925213 Simplifying n + 0.4166666667 = -0.600925213 Reorder the terms: 0.4166666667 + n = -0.600925213 Solving 0.4166666667 + n = -0.600925213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = -0.600925213 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = -0.600925213 + -0.4166666667 n = -0.600925213 + -0.4166666667 Combine like terms: -0.600925213 + -0.4166666667 = -1.0175918797 n = -1.0175918797 Simplifying n = -1.0175918797

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.1842585463, -1.0175918797}

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