5200=n(20-0.01n)+100

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Solution for 5200=n(20-0.01n)+100 equation:


Simplifying
5200 = n(20 + -0.01n) + 100
5200 = (20 * n + -0.01n * n) + 100
5200 = (20n + -0.01n2) + 100

Reorder the terms:
5200 = 100 + 20n + -0.01n2

Solving
5200 = 100 + 20n + -0.01n2

Solving for variable 'n'.

Combine like terms: 5200 + -100 = 5100
5100 + -20n + 0.01n2 = 100 + 20n + -0.01n2 + -100 + -20n + 0.01n2

Reorder the terms:
5100 + -20n + 0.01n2 = 100 + -100 + 20n + -20n + -0.01n2 + 0.01n2

Combine like terms: 100 + -100 = 0
5100 + -20n + 0.01n2 = 0 + 20n + -20n + -0.01n2 + 0.01n2
5100 + -20n + 0.01n2 = 20n + -20n + -0.01n2 + 0.01n2

Combine like terms: 20n + -20n = 0
5100 + -20n + 0.01n2 = 0 + -0.01n2 + 0.01n2
5100 + -20n + 0.01n2 = -0.01n2 + 0.01n2

Combine like terms: -0.01n2 + 0.01n2 = 0.00
5100 + -20n + 0.01n2 = 0.00

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

Divide each side by '0.01'.
510000 + -2000n + n2 = 0

Move the constant term to the right:

Add '-510000' to each side of the equation.
510000 + -2000n + -510000 + n2 = 0 + -510000

Reorder the terms:
510000 + -510000 + -2000n + n2 = 0 + -510000

Combine like terms: 510000 + -510000 = 0
0 + -2000n + n2 = 0 + -510000
-2000n + n2 = 0 + -510000

Combine like terms: 0 + -510000 = -510000
-2000n + n2 = -510000

The n term is -2000n.  Take half its coefficient (-1000).
Square it (1000000) and add it to both sides.

Add '1000000' to each side of the equation.
-2000n + 1000000 + n2 = -510000 + 1000000

Reorder the terms:
1000000 + -2000n + n2 = -510000 + 1000000

Combine like terms: -510000 + 1000000 = 490000
1000000 + -2000n + n2 = 490000

Factor a perfect square on the left side:
(n + -1000)(n + -1000) = 490000

Calculate the square root of the right side: 700

Break this problem into two subproblems by setting 
(n + -1000) equal to 700 and -700.

Subproblem 1

n + -1000 = 700 Simplifying n + -1000 = 700 Reorder the terms: -1000 + n = 700 Solving -1000 + n = 700 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + n = 700 + 1000 Combine like terms: -1000 + 1000 = 0 0 + n = 700 + 1000 n = 700 + 1000 Combine like terms: 700 + 1000 = 1700 n = 1700 Simplifying n = 1700

Subproblem 2

n + -1000 = -700 Simplifying n + -1000 = -700 Reorder the terms: -1000 + n = -700 Solving -1000 + n = -700 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1000' to each side of the equation. -1000 + 1000 + n = -700 + 1000 Combine like terms: -1000 + 1000 = 0 0 + n = -700 + 1000 n = -700 + 1000 Combine like terms: -700 + 1000 = 300 n = 300 Simplifying n = 300

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1700, 300}

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