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Simplifying 16 = 6.5n + -3.3n(2n + -5) Reorder the terms: 16 = 6.5n + -3.3n(-5 + 2n) 16 = 6.5n + (-5 * -3.3n + 2n * -3.3n) 16 = 6.5n + (16.5n + -6.6n2) Combine like terms: 6.5n + 16.5n = 23n 16 = 23n + -6.6n2 Solving 16 = 23n + -6.6n2 Solving for variable 'n'. Reorder the terms: 16 + -23n + 6.6n2 = 23n + -23n + -6.6n2 + 6.6n2 Combine like terms: 23n + -23n = 0 16 + -23n + 6.6n2 = 0 + -6.6n2 + 6.6n2 16 + -23n + 6.6n2 = -6.6n2 + 6.6n2 Combine like terms: -6.6n2 + 6.6n2 = 0.0 16 + -23n + 6.6n2 = 0.0 Begin completing the square. Divide all terms by 6.6 the coefficient of the squared term: Divide each side by '6.6'. 2.424242424 + -3.484848485n + n2 = 0 Move the constant term to the right: Add '-2.424242424' to each side of the equation. 2.424242424 + -3.484848485n + -2.424242424 + n2 = 0 + -2.424242424 Reorder the terms: 2.424242424 + -2.424242424 + -3.484848485n + n2 = 0 + -2.424242424 Combine like terms: 2.424242424 + -2.424242424 = 0.000000000 0.000000000 + -3.484848485n + n2 = 0 + -2.424242424 -3.484848485n + n2 = 0 + -2.424242424 Combine like terms: 0 + -2.424242424 = -2.424242424 -3.484848485n + n2 = -2.424242424 The n term is -3.484848485n. Take half its coefficient (-1.742424243). Square it (3.036042243) and add it to both sides. Add '3.036042243' to each side of the equation. -3.484848485n + 3.036042243 + n2 = -2.424242424 + 3.036042243 Reorder the terms: 3.036042243 + -3.484848485n + n2 = -2.424242424 + 3.036042243 Combine like terms: -2.424242424 + 3.036042243 = 0.611799819 3.036042243 + -3.484848485n + n2 = 0.611799819 Factor a perfect square on the left side: (n + -1.742424243)(n + -1.742424243) = 0.611799819 Calculate the square root of the right side: 0.782176335 Break this problem into two subproblems by setting (n + -1.742424243) equal to 0.782176335 and -0.782176335.Subproblem 1
n + -1.742424243 = 0.782176335 Simplifying n + -1.742424243 = 0.782176335 Reorder the terms: -1.742424243 + n = 0.782176335 Solving -1.742424243 + n = 0.782176335 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.742424243' to each side of the equation. -1.742424243 + 1.742424243 + n = 0.782176335 + 1.742424243 Combine like terms: -1.742424243 + 1.742424243 = 0.000000000 0.000000000 + n = 0.782176335 + 1.742424243 n = 0.782176335 + 1.742424243 Combine like terms: 0.782176335 + 1.742424243 = 2.524600578 n = 2.524600578 Simplifying n = 2.524600578Subproblem 2
n + -1.742424243 = -0.782176335 Simplifying n + -1.742424243 = -0.782176335 Reorder the terms: -1.742424243 + n = -0.782176335 Solving -1.742424243 + n = -0.782176335 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.742424243' to each side of the equation. -1.742424243 + 1.742424243 + n = -0.782176335 + 1.742424243 Combine like terms: -1.742424243 + 1.742424243 = 0.000000000 0.000000000 + n = -0.782176335 + 1.742424243 n = -0.782176335 + 1.742424243 Combine like terms: -0.782176335 + 1.742424243 = 0.960247908 n = 0.960247908 Simplifying n = 0.960247908Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.524600578, 0.960247908}
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