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