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Simplifying 10n2 + -18 = -9n Reorder the terms: -18 + 10n2 = -9n Solving -18 + 10n2 = -9n Solving for variable 'n'. Reorder the terms: -18 + 9n + 10n2 = -9n + 9n Combine like terms: -9n + 9n = 0 -18 + 9n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -1.8 + 0.9n + n2 = 0 Move the constant term to the right: Add '1.8' to each side of the equation. -1.8 + 0.9n + 1.8 + n2 = 0 + 1.8 Reorder the terms: -1.8 + 1.8 + 0.9n + n2 = 0 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + 0.9n + n2 = 0 + 1.8 0.9n + n2 = 0 + 1.8 Combine like terms: 0 + 1.8 = 1.8 0.9n + n2 = 1.8 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 = 1.8 + 0.2025 Reorder the terms: 0.2025 + 0.9n + n2 = 1.8 + 0.2025 Combine like terms: 1.8 + 0.2025 = 2.0025 0.2025 + 0.9n + n2 = 2.0025 Factor a perfect square on the left side: (n + 0.45)(n + 0.45) = 2.0025 Calculate the square root of the right side: 1.41509717 Break this problem into two subproblems by setting (n + 0.45) equal to 1.41509717 and -1.41509717.Subproblem 1
n + 0.45 = 1.41509717 Simplifying n + 0.45 = 1.41509717 Reorder the terms: 0.45 + n = 1.41509717 Solving 0.45 + n = 1.41509717 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 = 1.41509717 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = 1.41509717 + -0.45 n = 1.41509717 + -0.45 Combine like terms: 1.41509717 + -0.45 = 0.96509717 n = 0.96509717 Simplifying n = 0.96509717Subproblem 2
n + 0.45 = -1.41509717 Simplifying n + 0.45 = -1.41509717 Reorder the terms: 0.45 + n = -1.41509717 Solving 0.45 + n = -1.41509717 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 = -1.41509717 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + n = -1.41509717 + -0.45 n = -1.41509717 + -0.45 Combine like terms: -1.41509717 + -0.45 = -1.86509717 n = -1.86509717 Simplifying n = -1.86509717Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.96509717, -1.86509717}
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