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Simplifying 27 = 0.5n(n + -3) Reorder the terms: 27 = 0.5n(-3 + n) 27 = (-3 * 0.5n + n * 0.5n) 27 = (-1.5n + 0.5n2) Solving 27 = -1.5n + 0.5n2 Solving for variable 'n'. Reorder the terms: 27 + 1.5n + -0.5n2 = -1.5n + 1.5n + 0.5n2 + -0.5n2 Combine like terms: -1.5n + 1.5n = 0.0 27 + 1.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2 27 + 1.5n + -0.5n2 = 0.5n2 + -0.5n2 Combine like terms: 0.5n2 + -0.5n2 = 0.0 27 + 1.5n + -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'. -54 + -3n + n2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + -3n + 54 + n2 = 0 + 54 Reorder the terms: -54 + 54 + -3n + n2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -3n + n2 = 0 + 54 -3n + n2 = 0 + 54 Combine like terms: 0 + 54 = 54 -3n + n2 = 54 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 = 54 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 54 + 2.25 Combine like terms: 54 + 2.25 = 56.25 2.25 + -3n + n2 = 56.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (n + -1.5) equal to 7.5 and -7.5.Subproblem 1
n + -1.5 = 7.5 Simplifying n + -1.5 = 7.5 Reorder the terms: -1.5 + n = 7.5 Solving -1.5 + n = 7.5 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 = 7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 7.5 + 1.5 n = 7.5 + 1.5 Combine like terms: 7.5 + 1.5 = 9 n = 9 Simplifying n = 9Subproblem 2
n + -1.5 = -7.5 Simplifying n + -1.5 = -7.5 Reorder the terms: -1.5 + n = -7.5 Solving -1.5 + n = -7.5 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 = -7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -7.5 + 1.5 n = -7.5 + 1.5 Combine like terms: -7.5 + 1.5 = -6 n = -6 Simplifying n = -6Solution
The solution to the problem is based on the solutions from the subproblems. n = {9, -6}
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