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Simplifying n * 2.3(10 + -1n) * 2.2 = 32.9 Reorder the terms for easier multiplication: 2.3 * 2.2n(10 + -1n) = 32.9 Multiply 2.3 * 2.2 5.06n(10 + -1n) = 32.9 (10 * 5.06n + -1n * 5.06n) = 32.9 (50.6n + -5.06n2) = 32.9 Solving 50.6n + -5.06n2 = 32.9 Solving for variable 'n'. Reorder the terms: -32.9 + 50.6n + -5.06n2 = 32.9 + -32.9 Combine like terms: 32.9 + -32.9 = 0.0 -32.9 + 50.6n + -5.06n2 = 0.0 Begin completing the square. Divide all terms by -5.06 the coefficient of the squared term: Divide each side by '-5.06'. 6.501976285 + -10n + n2 = 0 Move the constant term to the right: Add '-6.501976285' to each side of the equation. 6.501976285 + -10n + -6.501976285 + n2 = 0 + -6.501976285 Reorder the terms: 6.501976285 + -6.501976285 + -10n + n2 = 0 + -6.501976285 Combine like terms: 6.501976285 + -6.501976285 = 0.000000000 0.000000000 + -10n + n2 = 0 + -6.501976285 -10n + n2 = 0 + -6.501976285 Combine like terms: 0 + -6.501976285 = -6.501976285 -10n + n2 = -6.501976285 The n term is -10n. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10n + 25 + n2 = -6.501976285 + 25 Reorder the terms: 25 + -10n + n2 = -6.501976285 + 25 Combine like terms: -6.501976285 + 25 = 18.498023715 25 + -10n + n2 = 18.498023715 Factor a perfect square on the left side: (n + -5)(n + -5) = 18.498023715 Calculate the square root of the right side: 4.300932889 Break this problem into two subproblems by setting (n + -5) equal to 4.300932889 and -4.300932889.Subproblem 1
n + -5 = 4.300932889 Simplifying n + -5 = 4.300932889 Reorder the terms: -5 + n = 4.300932889 Solving -5 + n = 4.300932889 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = 4.300932889 + 5 Combine like terms: -5 + 5 = 0 0 + n = 4.300932889 + 5 n = 4.300932889 + 5 Combine like terms: 4.300932889 + 5 = 9.300932889 n = 9.300932889 Simplifying n = 9.300932889Subproblem 2
n + -5 = -4.300932889 Simplifying n + -5 = -4.300932889 Reorder the terms: -5 + n = -4.300932889 Solving -5 + n = -4.300932889 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = -4.300932889 + 5 Combine like terms: -5 + 5 = 0 0 + n = -4.300932889 + 5 n = -4.300932889 + 5 Combine like terms: -4.300932889 + 5 = 0.699067111 n = 0.699067111 Simplifying n = 0.699067111Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.300932889, 0.699067111}
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