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Simplifying 9000n = -0.25n2 + 6n + -27 Reorder the terms: 9000n = -27 + 6n + -0.25n2 Solving 9000n = -27 + 6n + -0.25n2 Solving for variable 'n'. Reorder the terms: 27 + 9000n + -6n + 0.25n2 = -27 + 6n + -0.25n2 + 27 + -6n + 0.25n2 Combine like terms: 9000n + -6n = 8994n 27 + 8994n + 0.25n2 = -27 + 6n + -0.25n2 + 27 + -6n + 0.25n2 Reorder the terms: 27 + 8994n + 0.25n2 = -27 + 27 + 6n + -6n + -0.25n2 + 0.25n2 Combine like terms: -27 + 27 = 0 27 + 8994n + 0.25n2 = 0 + 6n + -6n + -0.25n2 + 0.25n2 27 + 8994n + 0.25n2 = 6n + -6n + -0.25n2 + 0.25n2 Combine like terms: 6n + -6n = 0 27 + 8994n + 0.25n2 = 0 + -0.25n2 + 0.25n2 27 + 8994n + 0.25n2 = -0.25n2 + 0.25n2 Combine like terms: -0.25n2 + 0.25n2 = 0.00 27 + 8994n + 0.25n2 = 0.00 Begin completing the square. Divide all terms by 0.25 the coefficient of the squared term: Divide each side by '0.25'. 108 + 35976n + n2 = 0 Move the constant term to the right: Add '-108' to each side of the equation. 108 + 35976n + -108 + n2 = 0 + -108 Reorder the terms: 108 + -108 + 35976n + n2 = 0 + -108 Combine like terms: 108 + -108 = 0 0 + 35976n + n2 = 0 + -108 35976n + n2 = 0 + -108 Combine like terms: 0 + -108 = -108 35976n + n2 = -108 The n term is 35976n. Take half its coefficient (17988). Square it (323568144) and add it to both sides. Add '323568144' to each side of the equation. 35976n + 323568144 + n2 = -108 + 323568144 Reorder the terms: 323568144 + 35976n + n2 = -108 + 323568144 Combine like terms: -108 + 323568144 = 323568036 323568144 + 35976n + n2 = 323568036 Factor a perfect square on the left side: (n + 17988)(n + 17988) = 323568036 Calculate the square root of the right side: 17987.996997998 Break this problem into two subproblems by setting (n + 17988) equal to 17987.996997998 and -17987.996997998.Subproblem 1
n + 17988 = 17987.996997998 Simplifying n + 17988 = 17987.996997998 Reorder the terms: 17988 + n = 17987.996997998 Solving 17988 + n = 17987.996997998 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-17988' to each side of the equation. 17988 + -17988 + n = 17987.996997998 + -17988 Combine like terms: 17988 + -17988 = 0 0 + n = 17987.996997998 + -17988 n = 17987.996997998 + -17988 Combine like terms: 17987.996997998 + -17988 = -0.003002002 n = -0.003002002 Simplifying n = -0.003002002Subproblem 2
n + 17988 = -17987.996997998 Simplifying n + 17988 = -17987.996997998 Reorder the terms: 17988 + n = -17987.996997998 Solving 17988 + n = -17987.996997998 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-17988' to each side of the equation. 17988 + -17988 + n = -17987.996997998 + -17988 Combine like terms: 17988 + -17988 = 0 0 + n = -17987.996997998 + -17988 n = -17987.996997998 + -17988 Combine like terms: -17987.996997998 + -17988 = -35975.996997998 n = -35975.996997998 Simplifying n = -35975.996997998Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.003002002, -35975.996997998}
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