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Simplifying n(n + -1) = 80 Reorder the terms: n(-1 + n) = 80 (-1 * n + n * n) = 80 (-1n + n2) = 80 Solving -1n + n2 = 80 Solving for variable 'n'. Reorder the terms: -80 + -1n + n2 = 80 + -80 Combine like terms: 80 + -80 = 0 -80 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '80' to each side of the equation. -80 + -1n + 80 + n2 = 0 + 80 Reorder the terms: -80 + 80 + -1n + n2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + -1n + n2 = 0 + 80 -1n + n2 = 0 + 80 Combine like terms: 0 + 80 = 80 -1n + n2 = 80 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 80 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 80 + 0.25 Combine like terms: 80 + 0.25 = 80.25 0.25 + -1n + n2 = 80.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 80.25 Calculate the square root of the right side: 8.958236434 Break this problem into two subproblems by setting (n + -0.5) equal to 8.958236434 and -8.958236434.Subproblem 1
n + -0.5 = 8.958236434 Simplifying n + -0.5 = 8.958236434 Reorder the terms: -0.5 + n = 8.958236434 Solving -0.5 + n = 8.958236434 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 8.958236434 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 8.958236434 + 0.5 n = 8.958236434 + 0.5 Combine like terms: 8.958236434 + 0.5 = 9.458236434 n = 9.458236434 Simplifying n = 9.458236434Subproblem 2
n + -0.5 = -8.958236434 Simplifying n + -0.5 = -8.958236434 Reorder the terms: -0.5 + n = -8.958236434 Solving -0.5 + n = -8.958236434 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -8.958236434 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -8.958236434 + 0.5 n = -8.958236434 + 0.5 Combine like terms: -8.958236434 + 0.5 = -8.458236434 n = -8.458236434 Simplifying n = -8.458236434Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.458236434, -8.458236434}
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