0.1n=25+0.05n(n-200)

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Solution for 0.1n=25+0.05n(n-200) equation:


Simplifying
0.1n = 25 + 0.05n(n + -200)

Reorder the terms:
0.1n = 25 + 0.05n(-200 + n)
0.1n = 25 + (-200 * 0.05n + n * 0.05n)
0.1n = 25 + (-10n + 0.05n2)

Solving
0.1n = 25 + -10n + 0.05n2

Solving for variable 'n'.

Reorder the terms:
-25 + 0.1n + 10n + -0.05n2 = 25 + -10n + 0.05n2 + -25 + 10n + -0.05n2

Combine like terms: 0.1n + 10n = 10.1n
-25 + 10.1n + -0.05n2 = 25 + -10n + 0.05n2 + -25 + 10n + -0.05n2

Reorder the terms:
-25 + 10.1n + -0.05n2 = 25 + -25 + -10n + 10n + 0.05n2 + -0.05n2

Combine like terms: 25 + -25 = 0
-25 + 10.1n + -0.05n2 = 0 + -10n + 10n + 0.05n2 + -0.05n2
-25 + 10.1n + -0.05n2 = -10n + 10n + 0.05n2 + -0.05n2

Combine like terms: -10n + 10n = 0
-25 + 10.1n + -0.05n2 = 0 + 0.05n2 + -0.05n2
-25 + 10.1n + -0.05n2 = 0.05n2 + -0.05n2

Combine like terms: 0.05n2 + -0.05n2 = 0.00
-25 + 10.1n + -0.05n2 = 0.00

Begin completing the square.  Divide all terms by
-0.05 the coefficient of the squared term: 

Divide each side by '-0.05'.
500 + -202n + n2 = 0

Move the constant term to the right:

Add '-500' to each side of the equation.
500 + -202n + -500 + n2 = 0 + -500

Reorder the terms:
500 + -500 + -202n + n2 = 0 + -500

Combine like terms: 500 + -500 = 0
0 + -202n + n2 = 0 + -500
-202n + n2 = 0 + -500

Combine like terms: 0 + -500 = -500
-202n + n2 = -500

The n term is -202n.  Take half its coefficient (-101).
Square it (10201) and add it to both sides.

Add '10201' to each side of the equation.
-202n + 10201 + n2 = -500 + 10201

Reorder the terms:
10201 + -202n + n2 = -500 + 10201

Combine like terms: -500 + 10201 = 9701
10201 + -202n + n2 = 9701

Factor a perfect square on the left side:
(n + -101)(n + -101) = 9701

Calculate the square root of the right side: 98.493654618

Break this problem into two subproblems by setting 
(n + -101) equal to 98.493654618 and -98.493654618.

Subproblem 1

n + -101 = 98.493654618 Simplifying n + -101 = 98.493654618 Reorder the terms: -101 + n = 98.493654618 Solving -101 + n = 98.493654618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '101' to each side of the equation. -101 + 101 + n = 98.493654618 + 101 Combine like terms: -101 + 101 = 0 0 + n = 98.493654618 + 101 n = 98.493654618 + 101 Combine like terms: 98.493654618 + 101 = 199.493654618 n = 199.493654618 Simplifying n = 199.493654618

Subproblem 2

n + -101 = -98.493654618 Simplifying n + -101 = -98.493654618 Reorder the terms: -101 + n = -98.493654618 Solving -101 + n = -98.493654618 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '101' to each side of the equation. -101 + 101 + n = -98.493654618 + 101 Combine like terms: -101 + 101 = 0 0 + n = -98.493654618 + 101 n = -98.493654618 + 101 Combine like terms: -98.493654618 + 101 = 2.506345382 n = 2.506345382 Simplifying n = 2.506345382

Solution

The solution to the problem is based on the solutions from the subproblems. n = {199.493654618, 2.506345382}

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