0.08n+.12n(n+500)=180

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Solution for 0.08n+.12n(n+500)=180 equation:


Simplifying
0.08n + 0.12n(n + 500) = 180

Reorder the terms:
0.08n + 0.12n(500 + n) = 180
0.08n + (500 * 0.12n + n * 0.12n) = 180
0.08n + (60n + 0.12n2) = 180

Combine like terms: 0.08n + 60n = 60.08n
60.08n + 0.12n2 = 180

Solving
60.08n + 0.12n2 = 180

Solving for variable 'n'.

Reorder the terms:
-180 + 60.08n + 0.12n2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 60.08n + 0.12n2 = 0

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

Divide each side by '0.12'.
-1500 + 500.6666667n + n2 = 0

Move the constant term to the right:

Add '1500' to each side of the equation.
-1500 + 500.6666667n + 1500 + n2 = 0 + 1500

Reorder the terms:
-1500 + 1500 + 500.6666667n + n2 = 0 + 1500

Combine like terms: -1500 + 1500 = 0
0 + 500.6666667n + n2 = 0 + 1500
500.6666667n + n2 = 0 + 1500

Combine like terms: 0 + 1500 = 1500
500.6666667n + n2 = 1500

The n term is 500.6666667n.  Take half its coefficient (250.3333334).
Square it (62666.77781) and add it to both sides.

Add '62666.77781' to each side of the equation.
500.6666667n + 62666.77781 + n2 = 1500 + 62666.77781

Reorder the terms:
62666.77781 + 500.6666667n + n2 = 1500 + 62666.77781

Combine like terms: 1500 + 62666.77781 = 64166.77781
62666.77781 + 500.6666667n + n2 = 64166.77781

Factor a perfect square on the left side:
(n + 250.3333334)(n + 250.3333334) = 64166.77781

Calculate the square root of the right side: 253.31162194

Break this problem into two subproblems by setting 
(n + 250.3333334) equal to 253.31162194 and -253.31162194.

Subproblem 1

n + 250.3333334 = 253.31162194 Simplifying n + 250.3333334 = 253.31162194 Reorder the terms: 250.3333334 + n = 253.31162194 Solving 250.3333334 + n = 253.31162194 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-250.3333334' to each side of the equation. 250.3333334 + -250.3333334 + n = 253.31162194 + -250.3333334 Combine like terms: 250.3333334 + -250.3333334 = 0.0000000 0.0000000 + n = 253.31162194 + -250.3333334 n = 253.31162194 + -250.3333334 Combine like terms: 253.31162194 + -250.3333334 = 2.97828854 n = 2.97828854 Simplifying n = 2.97828854

Subproblem 2

n + 250.3333334 = -253.31162194 Simplifying n + 250.3333334 = -253.31162194 Reorder the terms: 250.3333334 + n = -253.31162194 Solving 250.3333334 + n = -253.31162194 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-250.3333334' to each side of the equation. 250.3333334 + -250.3333334 + n = -253.31162194 + -250.3333334 Combine like terms: 250.3333334 + -250.3333334 = 0.0000000 0.0000000 + n = -253.31162194 + -250.3333334 n = -253.31162194 + -250.3333334 Combine like terms: -253.31162194 + -250.3333334 = -503.64495534 n = -503.64495534 Simplifying n = -503.64495534

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.97828854, -503.64495534}

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