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Simplifying 0.5n2 + 1.5n = 9 Reorder the terms: 1.5n + 0.5n2 = 9 Solving 1.5n + 0.5n2 = 9 Solving for variable 'n'. Reorder the terms: -9 + 1.5n + 0.5n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 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'. -18 + 3n + n2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + 3n + 18 + n2 = 0 + 18 Reorder the terms: -18 + 18 + 3n + n2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 3n + n2 = 0 + 18 3n + n2 = 0 + 18 Combine like terms: 0 + 18 = 18 3n + n2 = 18 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 = 18 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 18 + 2.25 Combine like terms: 18 + 2.25 = 20.25 2.25 + 3n + n2 = 20.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 20.25 Calculate the square root of the right side: 4.5 Break this problem into two subproblems by setting (n + 1.5) equal to 4.5 and -4.5.Subproblem 1
n + 1.5 = 4.5 Simplifying n + 1.5 = 4.5 Reorder the terms: 1.5 + n = 4.5 Solving 1.5 + n = 4.5 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 = 4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.5 + -1.5 n = 4.5 + -1.5 Combine like terms: 4.5 + -1.5 = 3 n = 3 Simplifying n = 3Subproblem 2
n + 1.5 = -4.5 Simplifying n + 1.5 = -4.5 Reorder the terms: 1.5 + n = -4.5 Solving 1.5 + n = -4.5 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 = -4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.5 + -1.5 n = -4.5 + -1.5 Combine like terms: -4.5 + -1.5 = -6 n = -6 Simplifying n = -6Solution
The solution to the problem is based on the solutions from the subproblems. n = {3, -6}
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