(3n-8)n=64

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Solution for (3n-8)n=64 equation:


Simplifying
(3n + -8) * n = 64

Reorder the terms:
(-8 + 3n) * n = 64

Reorder the terms for easier multiplication:
n(-8 + 3n) = 64
(-8 * n + 3n * n) = 64
(-8n + 3n2) = 64

Solving
-8n + 3n2 = 64

Solving for variable 'n'.

Reorder the terms:
-64 + -8n + 3n2 = 64 + -64

Combine like terms: 64 + -64 = 0
-64 + -8n + 3n2 = 0

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

Divide each side by '3'.
-21.33333333 + -2.666666667n + n2 = 0

Move the constant term to the right:

Add '21.33333333' to each side of the equation.
-21.33333333 + -2.666666667n + 21.33333333 + n2 = 0 + 21.33333333

Reorder the terms:
-21.33333333 + 21.33333333 + -2.666666667n + n2 = 0 + 21.33333333

Combine like terms: -21.33333333 + 21.33333333 = 0.00000000
0.00000000 + -2.666666667n + n2 = 0 + 21.33333333
-2.666666667n + n2 = 0 + 21.33333333

Combine like terms: 0 + 21.33333333 = 21.33333333
-2.666666667n + n2 = 21.33333333

The n term is -2.666666667n.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667n + 1.777777780 + n2 = 21.33333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667n + n2 = 21.33333333 + 1.777777780

Combine like terms: 21.33333333 + 1.777777780 = 23.11111111
1.777777780 + -2.666666667n + n2 = 23.11111111

Factor a perfect square on the left side:
(n + -1.333333334)(n + -1.333333334) = 23.11111111

Calculate the square root of the right side: 4.807401701

Break this problem into two subproblems by setting 
(n + -1.333333334) equal to 4.807401701 and -4.807401701.

Subproblem 1

n + -1.333333334 = 4.807401701 Simplifying n + -1.333333334 = 4.807401701 Reorder the terms: -1.333333334 + n = 4.807401701 Solving -1.333333334 + n = 4.807401701 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + n = 4.807401701 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + n = 4.807401701 + 1.333333334 n = 4.807401701 + 1.333333334 Combine like terms: 4.807401701 + 1.333333334 = 6.140735035 n = 6.140735035 Simplifying n = 6.140735035

Subproblem 2

n + -1.333333334 = -4.807401701 Simplifying n + -1.333333334 = -4.807401701 Reorder the terms: -1.333333334 + n = -4.807401701 Solving -1.333333334 + n = -4.807401701 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + n = -4.807401701 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + n = -4.807401701 + 1.333333334 n = -4.807401701 + 1.333333334 Combine like terms: -4.807401701 + 1.333333334 = -3.474068367 n = -3.474068367 Simplifying n = -3.474068367

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.140735035, -3.474068367}

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