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Simplifying 9n2 + 38n + 24 = 0 Reorder the terms: 24 + 38n + 9n2 = 0 Solving 24 + 38n + 9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 2.666666667 + 4.222222222n + n2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + 4.222222222n + -2.666666667 + n2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + 4.222222222n + n2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + 4.222222222n + n2 = 0 + -2.666666667 4.222222222n + n2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 4.222222222n + n2 = -2.666666667 The n term is 4.222222222n. Take half its coefficient (2.111111111). Square it (4.456790123) and add it to both sides. Add '4.456790123' to each side of the equation. 4.222222222n + 4.456790123 + n2 = -2.666666667 + 4.456790123 Reorder the terms: 4.456790123 + 4.222222222n + n2 = -2.666666667 + 4.456790123 Combine like terms: -2.666666667 + 4.456790123 = 1.790123456 4.456790123 + 4.222222222n + n2 = 1.790123456 Factor a perfect square on the left side: (n + 2.111111111)(n + 2.111111111) = 1.790123456 Calculate the square root of the right side: 1.337954953 Break this problem into two subproblems by setting (n + 2.111111111) equal to 1.337954953 and -1.337954953.Subproblem 1
n + 2.111111111 = 1.337954953 Simplifying n + 2.111111111 = 1.337954953 Reorder the terms: 2.111111111 + n = 1.337954953 Solving 2.111111111 + n = 1.337954953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.111111111' to each side of the equation. 2.111111111 + -2.111111111 + n = 1.337954953 + -2.111111111 Combine like terms: 2.111111111 + -2.111111111 = 0.000000000 0.000000000 + n = 1.337954953 + -2.111111111 n = 1.337954953 + -2.111111111 Combine like terms: 1.337954953 + -2.111111111 = -0.773156158 n = -0.773156158 Simplifying n = -0.773156158Subproblem 2
n + 2.111111111 = -1.337954953 Simplifying n + 2.111111111 = -1.337954953 Reorder the terms: 2.111111111 + n = -1.337954953 Solving 2.111111111 + n = -1.337954953 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.111111111' to each side of the equation. 2.111111111 + -2.111111111 + n = -1.337954953 + -2.111111111 Combine like terms: 2.111111111 + -2.111111111 = 0.000000000 0.000000000 + n = -1.337954953 + -2.111111111 n = -1.337954953 + -2.111111111 Combine like terms: -1.337954953 + -2.111111111 = -3.449066064 n = -3.449066064 Simplifying n = -3.449066064Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.773156158, -3.449066064}
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