2*n+n*3n=66

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


Simplifying
2n + n * 3n = 66

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

Multiply n * n
2n + 3n2 = 66

Solving
2n + 3n2 = 66

Solving for variable 'n'.

Reorder the terms:
-66 + 2n + 3n2 = 66 + -66

Combine like terms: 66 + -66 = 0
-66 + 2n + 3n2 = 0

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

Divide each side by '3'.
-22 + 0.6666666667n + n2 = 0

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 0.6666666667n + 22 + n2 = 0 + 22

Reorder the terms:
-22 + 22 + 0.6666666667n + n2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 0.6666666667n + n2 = 0 + 22
0.6666666667n + n2 = 0 + 22

Combine like terms: 0 + 22 = 22
0.6666666667n + n2 = 22

The n term is 0.6666666667n.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667n + 0.1111111112 + n2 = 22 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667n + n2 = 22 + 0.1111111112

Combine like terms: 22 + 0.1111111112 = 22.1111111112
0.1111111112 + 0.6666666667n + n2 = 22.1111111112

Factor a perfect square on the left side:
(n + 0.3333333334)(n + 0.3333333334) = 22.1111111112

Calculate the square root of the right side: 4.702245327

Break this problem into two subproblems by setting 
(n + 0.3333333334) equal to 4.702245327 and -4.702245327.

Subproblem 1

n + 0.3333333334 = 4.702245327 Simplifying n + 0.3333333334 = 4.702245327 Reorder the terms: 0.3333333334 + n = 4.702245327 Solving 0.3333333334 + n = 4.702245327 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = 4.702245327 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = 4.702245327 + -0.3333333334 n = 4.702245327 + -0.3333333334 Combine like terms: 4.702245327 + -0.3333333334 = 4.3689119936 n = 4.3689119936 Simplifying n = 4.3689119936

Subproblem 2

n + 0.3333333334 = -4.702245327 Simplifying n + 0.3333333334 = -4.702245327 Reorder the terms: 0.3333333334 + n = -4.702245327 Solving 0.3333333334 + n = -4.702245327 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + n = -4.702245327 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + n = -4.702245327 + -0.3333333334 n = -4.702245327 + -0.3333333334 Combine like terms: -4.702245327 + -0.3333333334 = -5.0355786604 n = -5.0355786604 Simplifying n = -5.0355786604

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.3689119936, -5.0355786604}

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