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Simplifying 0.5n(n + 1) = 66 Reorder the terms: 0.5n(1 + n) = 66 (1 * 0.5n + n * 0.5n) = 66 (0.5n + 0.5n2) = 66 Solving 0.5n + 0.5n2 = 66 Solving for variable 'n'. Reorder the terms: -66 + 0.5n + 0.5n2 = 66 + -66 Combine like terms: 66 + -66 = 0 -66 + 0.5n + 0.5n2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -132 + n + n2 = 0 Move the constant term to the right: Add '132' to each side of the equation. -132 + n + 132 + n2 = 0 + 132 Reorder the terms: -132 + 132 + n + n2 = 0 + 132 Combine like terms: -132 + 132 = 0 0 + n + n2 = 0 + 132 n + n2 = 0 + 132 Combine like terms: 0 + 132 = 132 n + n2 = 132 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 = 132 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 132 + 0.25 Combine like terms: 132 + 0.25 = 132.25 1.25 + n2 = 132.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 132.25 Calculate the square root of the right side: 11.5 Break this problem into two subproblems by setting (n + 0.5) equal to 11.5 and -11.5.Subproblem 1
n + 0.5 = 11.5 Simplifying n + 0.5 = 11.5 Reorder the terms: 0.5 + n = 11.5 Solving 0.5 + n = 11.5 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 = 11.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 11.5 + -0.5 n = 11.5 + -0.5 Combine like terms: 11.5 + -0.5 = 11 n = 11 Simplifying n = 11Subproblem 2
n + 0.5 = -11.5 Simplifying n + 0.5 = -11.5 Reorder the terms: 0.5 + n = -11.5 Solving 0.5 + n = -11.5 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 = -11.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -11.5 + -0.5 n = -11.5 + -0.5 Combine like terms: -11.5 + -0.5 = -12 n = -12 Simplifying n = -12Solution
The solution to the problem is based on the solutions from the subproblems. n = {11, -12}
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