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Simplifying 0.05n + 0.10n(2n + -1) = 1.40 Reorder the terms: 0.05n + 0.10n(-1 + 2n) = 1.40 0.05n + (-1 * 0.10n + 2n * 0.10n) = 1.40 0.05n + (-0.1n + 0.2n2) = 1.40 Combine like terms: 0.05n + -0.1n = -0.05n -0.05n + 0.2n2 = 1.40 Solving -0.05n + 0.2n2 = 1.40 Solving for variable 'n'. Reorder the terms: -1.40 + -0.05n + 0.2n2 = 1.40 + -1.40 Combine like terms: 1.40 + -1.40 = 0.00 -1.40 + -0.05n + 0.2n2 = 0.00 Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. -7 + -0.25n + n2 = 0.0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -0.25n + 7 + n2 = 0.0 + 7 Reorder the terms: -7 + 7 + -0.25n + n2 = 0.0 + 7 Combine like terms: -7 + 7 = 0 0 + -0.25n + n2 = 0.0 + 7 -0.25n + n2 = 0.0 + 7 Combine like terms: 0.0 + 7 = 7 -0.25n + n2 = 7 The n term is -0.25n. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25n + 0.015625 + n2 = 7 + 0.015625 Reorder the terms: 0.015625 + -0.25n + n2 = 7 + 0.015625 Combine like terms: 7 + 0.015625 = 7.015625 0.015625 + -0.25n + n2 = 7.015625 Factor a perfect square on the left side: (n + -0.125)(n + -0.125) = 7.015625 Calculate the square root of the right side: 2.648702513 Break this problem into two subproblems by setting (n + -0.125) equal to 2.648702513 and -2.648702513.Subproblem 1
n + -0.125 = 2.648702513 Simplifying n + -0.125 = 2.648702513 Reorder the terms: -0.125 + n = 2.648702513 Solving -0.125 + n = 2.648702513 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 2.648702513 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 2.648702513 + 0.125 n = 2.648702513 + 0.125 Combine like terms: 2.648702513 + 0.125 = 2.773702513 n = 2.773702513 Simplifying n = 2.773702513Subproblem 2
n + -0.125 = -2.648702513 Simplifying n + -0.125 = -2.648702513 Reorder the terms: -0.125 + n = -2.648702513 Solving -0.125 + n = -2.648702513 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -2.648702513 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -2.648702513 + 0.125 n = -2.648702513 + 0.125 Combine like terms: -2.648702513 + 0.125 = -2.523702513 n = -2.523702513 Simplifying n = -2.523702513Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.773702513, -2.523702513}
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