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Simplifying n(n + 1) = 750 Reorder the terms: n(1 + n) = 750 (1 * n + n * n) = 750 (1n + n2) = 750 Solving 1n + n2 = 750 Solving for variable 'n'. Reorder the terms: -750 + 1n + n2 = 750 + -750 Combine like terms: 750 + -750 = 0 -750 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '750' to each side of the equation. -750 + 1n + 750 + n2 = 0 + 750 Reorder the terms: -750 + 750 + 1n + n2 = 0 + 750 Combine like terms: -750 + 750 = 0 0 + 1n + n2 = 0 + 750 1n + n2 = 0 + 750 Combine like terms: 0 + 750 = 750 1n + n2 = 750 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. n + 0.25 + n2 = 750 + 0.25 Reorder the terms: 0.25 + n + n2 = 750 + 0.25 Combine like terms: 750 + 0.25 = 750.25 0.25 + n + n2 = 750.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 750.25 Calculate the square root of the right side: 27.39069185 Break this problem into two subproblems by setting (n + 0.5) equal to 27.39069185 and -27.39069185.Subproblem 1
n + 0.5 = 27.39069185 Simplifying n + 0.5 = 27.39069185 Reorder the terms: 0.5 + n = 27.39069185 Solving 0.5 + n = 27.39069185 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 = 27.39069185 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 27.39069185 + -0.5 n = 27.39069185 + -0.5 Combine like terms: 27.39069185 + -0.5 = 26.89069185 n = 26.89069185 Simplifying n = 26.89069185Subproblem 2
n + 0.5 = -27.39069185 Simplifying n + 0.5 = -27.39069185 Reorder the terms: 0.5 + n = -27.39069185 Solving 0.5 + n = -27.39069185 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 = -27.39069185 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -27.39069185 + -0.5 n = -27.39069185 + -0.5 Combine like terms: -27.39069185 + -0.5 = -27.89069185 n = -27.89069185 Simplifying n = -27.89069185Solution
The solution to the problem is based on the solutions from the subproblems. n = {26.89069185, -27.89069185}
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