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