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Simplifying 12 = 0.5n(n + -1) Reorder the terms: 12 = 0.5n(-1 + n) 12 = (-1 * 0.5n + n * 0.5n) 12 = (-0.5n + 0.5n2) Solving 12 = -0.5n + 0.5n2 Solving for variable 'n'. Reorder the terms: 12 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2 Combine like terms: -0.5n + 0.5n = 0.0 12 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2 12 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2 Combine like terms: 0.5n2 + -0.5n2 = 0.0 12 + 0.5n + -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'. -24 + -1n + n2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + -1n + 24 + n2 = 0 + 24 Reorder the terms: -24 + 24 + -1n + n2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -1n + n2 = 0 + 24 -1n + n2 = 0 + 24 Combine like terms: 0 + 24 = 24 -1n + n2 = 24 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 = 24 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 24 + 0.25 Combine like terms: 24 + 0.25 = 24.25 0.25 + -1n + n2 = 24.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (n + -0.5) equal to 4.924428901 and -4.924428901.Subproblem 1
n + -0.5 = 4.924428901 Simplifying n + -0.5 = 4.924428901 Reorder the terms: -0.5 + n = 4.924428901 Solving -0.5 + n = 4.924428901 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 = 4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 4.924428901 + 0.5 n = 4.924428901 + 0.5 Combine like terms: 4.924428901 + 0.5 = 5.424428901 n = 5.424428901 Simplifying n = 5.424428901Subproblem 2
n + -0.5 = -4.924428901 Simplifying n + -0.5 = -4.924428901 Reorder the terms: -0.5 + n = -4.924428901 Solving -0.5 + n = -4.924428901 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 = -4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -4.924428901 + 0.5 n = -4.924428901 + 0.5 Combine like terms: -4.924428901 + 0.5 = -4.424428901 n = -4.424428901 Simplifying n = -4.424428901Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.424428901, -4.424428901}
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