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