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Simplifying 5n2 + -32n + 16 = 0 Reorder the terms: 16 + -32n + 5n2 = 0 Solving 16 + -32n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 3.2 + -6.4n + n2 = 0 Move the constant term to the right: Add '-3.2' to each side of the equation. 3.2 + -6.4n + -3.2 + n2 = 0 + -3.2 Reorder the terms: 3.2 + -3.2 + -6.4n + n2 = 0 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + -6.4n + n2 = 0 + -3.2 -6.4n + n2 = 0 + -3.2 Combine like terms: 0 + -3.2 = -3.2 -6.4n + n2 = -3.2 The n term is -6.4n. Take half its coefficient (-3.2). Square it (10.24) and add it to both sides. Add '10.24' to each side of the equation. -6.4n + 10.24 + n2 = -3.2 + 10.24 Reorder the terms: 10.24 + -6.4n + n2 = -3.2 + 10.24 Combine like terms: -3.2 + 10.24 = 7.04 10.24 + -6.4n + n2 = 7.04 Factor a perfect square on the left side: (n + -3.2)(n + -3.2) = 7.04 Calculate the square root of the right side: 2.653299832 Break this problem into two subproblems by setting (n + -3.2) equal to 2.653299832 and -2.653299832.Subproblem 1
n + -3.2 = 2.653299832 Simplifying n + -3.2 = 2.653299832 Reorder the terms: -3.2 + n = 2.653299832 Solving -3.2 + n = 2.653299832 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = 2.653299832 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = 2.653299832 + 3.2 n = 2.653299832 + 3.2 Combine like terms: 2.653299832 + 3.2 = 5.853299832 n = 5.853299832 Simplifying n = 5.853299832Subproblem 2
n + -3.2 = -2.653299832 Simplifying n + -3.2 = -2.653299832 Reorder the terms: -3.2 + n = -2.653299832 Solving -3.2 + n = -2.653299832 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = -2.653299832 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = -2.653299832 + 3.2 n = -2.653299832 + 3.2 Combine like terms: -2.653299832 + 3.2 = 0.546700168 n = 0.546700168 Simplifying n = 0.546700168Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.853299832, 0.546700168}
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