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Simplifying 0.5n(n + -3) = 20 Reorder the terms: 0.5n(-3 + n) = 20 (-3 * 0.5n + n * 0.5n) = 20 (-1.5n + 0.5n2) = 20 Solving -1.5n + 0.5n2 = 20 Solving for variable 'n'. Reorder the terms: -20 + -1.5n + 0.5n2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + -1.5n + 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'. -40 + -3n + n2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + -3n + 40 + n2 = 0 + 40 Reorder the terms: -40 + 40 + -3n + n2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + -3n + n2 = 0 + 40 -3n + n2 = 0 + 40 Combine like terms: 0 + 40 = 40 -3n + n2 = 40 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 40 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 40 + 2.25 Combine like terms: 40 + 2.25 = 42.25 2.25 + -3n + n2 = 42.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (n + -1.5) equal to 6.5 and -6.5.Subproblem 1
n + -1.5 = 6.5 Simplifying n + -1.5 = 6.5 Reorder the terms: -1.5 + n = 6.5 Solving -1.5 + n = 6.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 6.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 6.5 + 1.5 n = 6.5 + 1.5 Combine like terms: 6.5 + 1.5 = 8 n = 8 Simplifying n = 8Subproblem 2
n + -1.5 = -6.5 Simplifying n + -1.5 = -6.5 Reorder the terms: -1.5 + n = -6.5 Solving -1.5 + n = -6.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -6.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -6.5 + 1.5 n = -6.5 + 1.5 Combine like terms: -6.5 + 1.5 = -5 n = -5 Simplifying n = -5Solution
The solution to the problem is based on the solutions from the subproblems. n = {8, -5}
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