8n(6n+7)=3

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Solution for 8n(6n+7)=3 equation:


Simplifying
8n(6n + 7) = 3

Reorder the terms:
8n(7 + 6n) = 3
(7 * 8n + 6n * 8n) = 3
(56n + 48n2) = 3

Solving
56n + 48n2 = 3

Solving for variable 'n'.

Reorder the terms:
-3 + 56n + 48n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 56n + 48n2 = 0

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

Divide each side by '48'.
-0.0625 + 1.166666667n + n2 = 0

Move the constant term to the right:

Add '0.0625' to each side of the equation.
-0.0625 + 1.166666667n + 0.0625 + n2 = 0 + 0.0625

Reorder the terms:
-0.0625 + 0.0625 + 1.166666667n + n2 = 0 + 0.0625

Combine like terms: -0.0625 + 0.0625 = 0.0000
0.0000 + 1.166666667n + n2 = 0 + 0.0625
1.166666667n + n2 = 0 + 0.0625

Combine like terms: 0 + 0.0625 = 0.0625
1.166666667n + n2 = 0.0625

The n term is 1.166666667n.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667n + 0.3402777780 + n2 = 0.0625 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667n + n2 = 0.0625 + 0.3402777780

Combine like terms: 0.0625 + 0.3402777780 = 0.402777778
0.3402777780 + 1.166666667n + n2 = 0.402777778

Factor a perfect square on the left side:
(n + 0.5833333335)(n + 0.5833333335) = 0.402777778

Calculate the square root of the right side: 0.634647759

Break this problem into two subproblems by setting 
(n + 0.5833333335) equal to 0.634647759 and -0.634647759.

Subproblem 1

n + 0.5833333335 = 0.634647759 Simplifying n + 0.5833333335 = 0.634647759 Reorder the terms: 0.5833333335 + n = 0.634647759 Solving 0.5833333335 + n = 0.634647759 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = 0.634647759 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = 0.634647759 + -0.5833333335 n = 0.634647759 + -0.5833333335 Combine like terms: 0.634647759 + -0.5833333335 = 0.0513144255 n = 0.0513144255 Simplifying n = 0.0513144255

Subproblem 2

n + 0.5833333335 = -0.634647759 Simplifying n + 0.5833333335 = -0.634647759 Reorder the terms: 0.5833333335 + n = -0.634647759 Solving 0.5833333335 + n = -0.634647759 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = -0.634647759 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = -0.634647759 + -0.5833333335 n = -0.634647759 + -0.5833333335 Combine like terms: -0.634647759 + -0.5833333335 = -1.2179810925 n = -1.2179810925 Simplifying n = -1.2179810925

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.0513144255, -1.2179810925}

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