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Simplifying 7264 = n(7n + 8) Reorder the terms: 7264 = n(8 + 7n) 7264 = (8 * n + 7n * n) 7264 = (8n + 7n2) Solving 7264 = 8n + 7n2 Solving for variable 'n'. Reorder the terms: 7264 + -8n + -7n2 = 8n + -8n + 7n2 + -7n2 Combine like terms: 8n + -8n = 0 7264 + -8n + -7n2 = 0 + 7n2 + -7n2 7264 + -8n + -7n2 = 7n2 + -7n2 Combine like terms: 7n2 + -7n2 = 0 7264 + -8n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1037.714286 + 1.142857143n + n2 = 0 Move the constant term to the right: Add '1037.714286' to each side of the equation. -1037.714286 + 1.142857143n + 1037.714286 + n2 = 0 + 1037.714286 Reorder the terms: -1037.714286 + 1037.714286 + 1.142857143n + n2 = 0 + 1037.714286 Combine like terms: -1037.714286 + 1037.714286 = 0.000000 0.000000 + 1.142857143n + n2 = 0 + 1037.714286 1.142857143n + n2 = 0 + 1037.714286 Combine like terms: 0 + 1037.714286 = 1037.714286 1.142857143n + n2 = 1037.714286 The n term is 1.142857143n. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143n + 0.3265306123 + n2 = 1037.714286 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143n + n2 = 1037.714286 + 0.3265306123 Combine like terms: 1037.714286 + 0.3265306123 = 1038.0408166123 0.3265306123 + 1.142857143n + n2 = 1038.0408166123 Factor a perfect square on the left side: (n + 0.5714285715)(n + 0.5714285715) = 1038.0408166123 Calculate the square root of the right side: 32.218640825 Break this problem into two subproblems by setting (n + 0.5714285715) equal to 32.218640825 and -32.218640825.Subproblem 1
n + 0.5714285715 = 32.218640825 Simplifying n + 0.5714285715 = 32.218640825 Reorder the terms: 0.5714285715 + n = 32.218640825 Solving 0.5714285715 + n = 32.218640825 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + n = 32.218640825 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + n = 32.218640825 + -0.5714285715 n = 32.218640825 + -0.5714285715 Combine like terms: 32.218640825 + -0.5714285715 = 31.6472122535 n = 31.6472122535 Simplifying n = 31.6472122535Subproblem 2
n + 0.5714285715 = -32.218640825 Simplifying n + 0.5714285715 = -32.218640825 Reorder the terms: 0.5714285715 + n = -32.218640825 Solving 0.5714285715 + n = -32.218640825 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + n = -32.218640825 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + n = -32.218640825 + -0.5714285715 n = -32.218640825 + -0.5714285715 Combine like terms: -32.218640825 + -0.5714285715 = -32.7900693965 n = -32.7900693965 Simplifying n = -32.7900693965Solution
The solution to the problem is based on the solutions from the subproblems. n = {31.6472122535, -32.7900693965}
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