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Simplifying 1275 = 0.5n(1 + n) 1275 = (1 * 0.5n + n * 0.5n) 1275 = (0.5n + 0.5n2) Solving 1275 = 0.5n + 0.5n2 Solving for variable 'n'. Reorder the terms: 1275 + -0.5n + -0.5n2 = 0.5n + -0.5n + 0.5n2 + -0.5n2 Combine like terms: 0.5n + -0.5n = 0.0 1275 + -0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2 1275 + -0.5n + -0.5n2 = 0.5n2 + -0.5n2 Combine like terms: 0.5n2 + -0.5n2 = 0.0 1275 + -0.5n + -0.5n2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -2550 + n + n2 = 0 Move the constant term to the right: Add '2550' to each side of the equation. -2550 + n + 2550 + n2 = 0 + 2550 Reorder the terms: -2550 + 2550 + n + n2 = 0 + 2550 Combine like terms: -2550 + 2550 = 0 0 + n + n2 = 0 + 2550 n + n2 = 0 + 2550 Combine like terms: 0 + 2550 = 2550 n + n2 = 2550 The n term is n. 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. + 0.25 + n2 = 2550 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 2550 + 0.25 Combine like terms: 2550 + 0.25 = 2550.25 1.25 + n2 = 2550.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 2550.25 Calculate the square root of the right side: 50.5 Break this problem into two subproblems by setting (n + 0.5) equal to 50.5 and -50.5.Subproblem 1
n + 0.5 = 50.5 Simplifying n + 0.5 = 50.5 Reorder the terms: 0.5 + n = 50.5 Solving 0.5 + n = 50.5 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 = 50.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 50.5 + -0.5 n = 50.5 + -0.5 Combine like terms: 50.5 + -0.5 = 50 n = 50 Simplifying n = 50Subproblem 2
n + 0.5 = -50.5 Simplifying n + 0.5 = -50.5 Reorder the terms: 0.5 + n = -50.5 Solving 0.5 + n = -50.5 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 = -50.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -50.5 + -0.5 n = -50.5 + -0.5 Combine like terms: -50.5 + -0.5 = -51 n = -51 Simplifying n = -51Solution
The solution to the problem is based on the solutions from the subproblems. n = {50, -51}
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