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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 + -1n + n2 = 0 Move the constant term to the right: Add '132' to each side of the equation. -132 + -1n + 132 + n2 = 0 + 132 Reorder the terms: -132 + 132 + -1n + n2 = 0 + 132 Combine like terms: -132 + 132 = 0 0 + -1n + n2 = 0 + 132 -1n + n2 = 0 + 132 Combine like terms: 0 + 132 = 132 -1n + n2 = 132 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 = 132 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 132 + 0.25 Combine like terms: 132 + 0.25 = 132.25 0.25 + -1n + 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 = 12 n = 12 Simplifying n = 12Subproblem 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 = -11 n = -11 Simplifying n = -11Solution
The solution to the problem is based on the solutions from the subproblems. n = {12, -11}
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