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Simplifying 24n2 + 49n + -96 = 0 Reorder the terms: -96 + 49n + 24n2 = 0 Solving -96 + 49n + 24n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -4 + 2.041666667n + n2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.041666667n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + 2.041666667n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.041666667n + n2 = 0 + 4 2.041666667n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.041666667n + n2 = 4 The n term is 2.041666667n. Take half its coefficient (1.020833334). Square it (1.042100696) and add it to both sides. Add '1.042100696' to each side of the equation. 2.041666667n + 1.042100696 + n2 = 4 + 1.042100696 Reorder the terms: 1.042100696 + 2.041666667n + n2 = 4 + 1.042100696 Combine like terms: 4 + 1.042100696 = 5.042100696 1.042100696 + 2.041666667n + n2 = 5.042100696 Factor a perfect square on the left side: (n + 1.020833334)(n + 1.020833334) = 5.042100696 Calculate the square root of the right side: 2.245462246 Break this problem into two subproblems by setting (n + 1.020833334) equal to 2.245462246 and -2.245462246.Subproblem 1
n + 1.020833334 = 2.245462246 Simplifying n + 1.020833334 = 2.245462246 Reorder the terms: 1.020833334 + n = 2.245462246 Solving 1.020833334 + n = 2.245462246 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + n = 2.245462246 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + n = 2.245462246 + -1.020833334 n = 2.245462246 + -1.020833334 Combine like terms: 2.245462246 + -1.020833334 = 1.224628912 n = 1.224628912 Simplifying n = 1.224628912Subproblem 2
n + 1.020833334 = -2.245462246 Simplifying n + 1.020833334 = -2.245462246 Reorder the terms: 1.020833334 + n = -2.245462246 Solving 1.020833334 + n = -2.245462246 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + n = -2.245462246 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + n = -2.245462246 + -1.020833334 n = -2.245462246 + -1.020833334 Combine like terms: -2.245462246 + -1.020833334 = -3.26629558 n = -3.26629558 Simplifying n = -3.26629558Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.224628912, -3.26629558}
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