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Simplifying n2 + 5n + -65 = 0 Reorder the terms: -65 + 5n + n2 = 0 Solving -65 + 5n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '65' to each side of the equation. -65 + 5n + 65 + n2 = 0 + 65 Reorder the terms: -65 + 65 + 5n + n2 = 0 + 65 Combine like terms: -65 + 65 = 0 0 + 5n + n2 = 0 + 65 5n + n2 = 0 + 65 Combine like terms: 0 + 65 = 65 5n + n2 = 65 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 65 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 65 + 6.25 Combine like terms: 65 + 6.25 = 71.25 6.25 + 5n + n2 = 71.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 71.25 Calculate the square root of the right side: 8.440971508 Break this problem into two subproblems by setting (n + 2.5) equal to 8.440971508 and -8.440971508.Subproblem 1
n + 2.5 = 8.440971508 Simplifying n + 2.5 = 8.440971508 Reorder the terms: 2.5 + n = 8.440971508 Solving 2.5 + n = 8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 8.440971508 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 8.440971508 + -2.5 n = 8.440971508 + -2.5 Combine like terms: 8.440971508 + -2.5 = 5.940971508 n = 5.940971508 Simplifying n = 5.940971508Subproblem 2
n + 2.5 = -8.440971508 Simplifying n + 2.5 = -8.440971508 Reorder the terms: 2.5 + n = -8.440971508 Solving 2.5 + n = -8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -8.440971508 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -8.440971508 + -2.5 n = -8.440971508 + -2.5 Combine like terms: -8.440971508 + -2.5 = -10.940971508 n = -10.940971508 Simplifying n = -10.940971508Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.940971508, -10.940971508}
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