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Simplifying n(n + -3) = 65 Reorder the terms: n(-3 + n) = 65 (-3 * n + n * n) = 65 (-3n + n2) = 65 Solving -3n + n2 = 65 Solving for variable 'n'. Reorder the terms: -65 + -3n + n2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '65' to each side of the equation. -65 + -3n + 65 + n2 = 0 + 65 Reorder the terms: -65 + 65 + -3n + n2 = 0 + 65 Combine like terms: -65 + 65 = 0 0 + -3n + n2 = 0 + 65 -3n + n2 = 0 + 65 Combine like terms: 0 + 65 = 65 -3n + n2 = 65 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 65 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 65 + 2.25 Combine like terms: 65 + 2.25 = 67.25 2.25 + -3n + n2 = 67.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 67.25 Calculate the square root of the right side: 8.200609733 Break this problem into two subproblems by setting (n + -1.5) equal to 8.200609733 and -8.200609733.Subproblem 1
n + -1.5 = 8.200609733 Simplifying n + -1.5 = 8.200609733 Reorder the terms: -1.5 + n = 8.200609733 Solving -1.5 + n = 8.200609733 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 8.200609733 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 8.200609733 + 1.5 n = 8.200609733 + 1.5 Combine like terms: 8.200609733 + 1.5 = 9.700609733 n = 9.700609733 Simplifying n = 9.700609733Subproblem 2
n + -1.5 = -8.200609733 Simplifying n + -1.5 = -8.200609733 Reorder the terms: -1.5 + n = -8.200609733 Solving -1.5 + n = -8.200609733 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -8.200609733 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -8.200609733 + 1.5 n = -8.200609733 + 1.5 Combine like terms: -8.200609733 + 1.5 = -6.700609733 n = -6.700609733 Simplifying n = -6.700609733Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.700609733, -6.700609733}
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