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Simplifying 7n2 + 488n + -5 = 7n + 1 Reorder the terms: -5 + 488n + 7n2 = 7n + 1 Reorder the terms: -5 + 488n + 7n2 = 1 + 7n Solving -5 + 488n + 7n2 = 1 + 7n Solving for variable 'n'. Reorder the terms: -5 + -1 + 488n + -7n + 7n2 = 1 + 7n + -1 + -7n Combine like terms: -5 + -1 = -6 -6 + 488n + -7n + 7n2 = 1 + 7n + -1 + -7n Combine like terms: 488n + -7n = 481n -6 + 481n + 7n2 = 1 + 7n + -1 + -7n Reorder the terms: -6 + 481n + 7n2 = 1 + -1 + 7n + -7n Combine like terms: 1 + -1 = 0 -6 + 481n + 7n2 = 0 + 7n + -7n -6 + 481n + 7n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -6 + 481n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + 68.71428571n + n2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + 68.71428571n + 0.8571428571 + n2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + 68.71428571n + n2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + 68.71428571n + n2 = 0 + 0.8571428571 68.71428571n + n2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 68.71428571n + n2 = 0.8571428571 The n term is 68.71428571n. Take half its coefficient (34.35714286). Square it (1180.413266) and add it to both sides. Add '1180.413266' to each side of the equation. 68.71428571n + 1180.413266 + n2 = 0.8571428571 + 1180.413266 Reorder the terms: 1180.413266 + 68.71428571n + n2 = 0.8571428571 + 1180.413266 Combine like terms: 0.8571428571 + 1180.413266 = 1181.2704088571 1180.413266 + 68.71428571n + n2 = 1181.2704088571 Factor a perfect square on the left side: (n + 34.35714286)(n + 34.35714286) = 1181.2704088571 Calculate the square root of the right side: 34.369614616 Break this problem into two subproblems by setting (n + 34.35714286) equal to 34.369614616 and -34.369614616.Subproblem 1
n + 34.35714286 = 34.369614616 Simplifying n + 34.35714286 = 34.369614616 Reorder the terms: 34.35714286 + n = 34.369614616 Solving 34.35714286 + n = 34.369614616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-34.35714286' to each side of the equation. 34.35714286 + -34.35714286 + n = 34.369614616 + -34.35714286 Combine like terms: 34.35714286 + -34.35714286 = 0.00000000 0.00000000 + n = 34.369614616 + -34.35714286 n = 34.369614616 + -34.35714286 Combine like terms: 34.369614616 + -34.35714286 = 0.012471756 n = 0.012471756 Simplifying n = 0.012471756Subproblem 2
n + 34.35714286 = -34.369614616 Simplifying n + 34.35714286 = -34.369614616 Reorder the terms: 34.35714286 + n = -34.369614616 Solving 34.35714286 + n = -34.369614616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-34.35714286' to each side of the equation. 34.35714286 + -34.35714286 + n = -34.369614616 + -34.35714286 Combine like terms: 34.35714286 + -34.35714286 = 0.00000000 0.00000000 + n = -34.369614616 + -34.35714286 n = -34.369614616 + -34.35714286 Combine like terms: -34.369614616 + -34.35714286 = -68.726757476 n = -68.726757476 Simplifying n = -68.726757476Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.012471756, -68.726757476}
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