5n+2n*3n=66

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Solution for 5n+2n*3n=66 equation:


Simplifying
5n + 2n * 3n = 66

Reorder the terms for easier multiplication:
5n + 2 * 3n * n = 66

Multiply 2 * 3
5n + 6n * n = 66

Multiply n * n
5n + 6n2 = 66

Solving
5n + 6n2 = 66

Solving for variable 'n'.

Reorder the terms:
-66 + 5n + 6n2 = 66 + -66

Combine like terms: 66 + -66 = 0
-66 + 5n + 6n2 = 0

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

Divide each side by '6'.
-11 + 0.8333333333n + n2 = 0

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 0.8333333333n + 11 + n2 = 0 + 11

Reorder the terms:
-11 + 11 + 0.8333333333n + n2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 0.8333333333n + n2 = 0 + 11
0.8333333333n + n2 = 0 + 11

Combine like terms: 0 + 11 = 11
0.8333333333n + n2 = 11

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 = 11 + 0.1736111111

Reorder the terms:
0.1736111111 + 0.8333333333n + n2 = 11 + 0.1736111111

Combine like terms: 11 + 0.1736111111 = 11.1736111111
0.1736111111 + 0.8333333333n + n2 = 11.1736111111

Factor a perfect square on the left side:
(n + 0.4166666667)(n + 0.4166666667) = 11.1736111111

Calculate the square root of the right side: 3.342695187

Break this problem into two subproblems by setting 
(n + 0.4166666667) equal to 3.342695187 and -3.342695187.

Subproblem 1

n + 0.4166666667 = 3.342695187 Simplifying n + 0.4166666667 = 3.342695187 Reorder the terms: 0.4166666667 + n = 3.342695187 Solving 0.4166666667 + n = 3.342695187 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 = 3.342695187 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = 3.342695187 + -0.4166666667 n = 3.342695187 + -0.4166666667 Combine like terms: 3.342695187 + -0.4166666667 = 2.9260285203 n = 2.9260285203 Simplifying n = 2.9260285203

Subproblem 2

n + 0.4166666667 = -3.342695187 Simplifying n + 0.4166666667 = -3.342695187 Reorder the terms: 0.4166666667 + n = -3.342695187 Solving 0.4166666667 + n = -3.342695187 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 = -3.342695187 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = -3.342695187 + -0.4166666667 n = -3.342695187 + -0.4166666667 Combine like terms: -3.342695187 + -0.4166666667 = -3.7593618537 n = -3.7593618537 Simplifying n = -3.7593618537

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.9260285203, -3.7593618537}

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