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Simplifying -10n2 + 9n + 24 = 0 Reorder the terms: 24 + 9n + -10n2 = 0 Solving 24 + 9n + -10n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -2.4 + -0.9n + n2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + -0.9n + 2.4 + n2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + -0.9n + n2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + -0.9n + n2 = 0 + 2.4 -0.9n + n2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 -0.9n + n2 = 2.4 The n term is -0.9n. Take half its coefficient (-0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. -0.9n + 0.2025 + n2 = 2.4 + 0.2025 Reorder the terms: 0.2025 + -0.9n + n2 = 2.4 + 0.2025 Combine like terms: 2.4 + 0.2025 = 2.6025 0.2025 + -0.9n + n2 = 2.6025 Factor a perfect square on the left side: (n + -0.45)(n + -0.45) = 2.6025 Calculate the square root of the right side: 1.61322658 Break this problem into two subproblems by setting (n + -0.45) equal to 1.61322658 and -1.61322658.Subproblem 1
n + -0.45 = 1.61322658 Simplifying n + -0.45 = 1.61322658 Reorder the terms: -0.45 + n = 1.61322658 Solving -0.45 + n = 1.61322658 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + n = 1.61322658 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + n = 1.61322658 + 0.45 n = 1.61322658 + 0.45 Combine like terms: 1.61322658 + 0.45 = 2.06322658 n = 2.06322658 Simplifying n = 2.06322658Subproblem 2
n + -0.45 = -1.61322658 Simplifying n + -0.45 = -1.61322658 Reorder the terms: -0.45 + n = -1.61322658 Solving -0.45 + n = -1.61322658 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + n = -1.61322658 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + n = -1.61322658 + 0.45 n = -1.61322658 + 0.45 Combine like terms: -1.61322658 + 0.45 = -1.16322658 n = -1.16322658 Simplifying n = -1.16322658Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.06322658, -1.16322658}
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