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Simplifying 7n2 + 51n + 62 = 0 Reorder the terms: 62 + 51n + 7n2 = 0 Solving 62 + 51n + 7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 8.857142857 + 7.285714286n + n2 = 0 Move the constant term to the right: Add '-8.857142857' to each side of the equation. 8.857142857 + 7.285714286n + -8.857142857 + n2 = 0 + -8.857142857 Reorder the terms: 8.857142857 + -8.857142857 + 7.285714286n + n2 = 0 + -8.857142857 Combine like terms: 8.857142857 + -8.857142857 = 0.000000000 0.000000000 + 7.285714286n + n2 = 0 + -8.857142857 7.285714286n + n2 = 0 + -8.857142857 Combine like terms: 0 + -8.857142857 = -8.857142857 7.285714286n + n2 = -8.857142857 The n term is 7.285714286n. Take half its coefficient (3.642857143). Square it (13.27040816) and add it to both sides. Add '13.27040816' to each side of the equation. 7.285714286n + 13.27040816 + n2 = -8.857142857 + 13.27040816 Reorder the terms: 13.27040816 + 7.285714286n + n2 = -8.857142857 + 13.27040816 Combine like terms: -8.857142857 + 13.27040816 = 4.413265303 13.27040816 + 7.285714286n + n2 = 4.413265303 Factor a perfect square on the left side: (n + 3.642857143)(n + 3.642857143) = 4.413265303 Calculate the square root of the right side: 2.100777309 Break this problem into two subproblems by setting (n + 3.642857143) equal to 2.100777309 and -2.100777309.Subproblem 1
n + 3.642857143 = 2.100777309 Simplifying n + 3.642857143 = 2.100777309 Reorder the terms: 3.642857143 + n = 2.100777309 Solving 3.642857143 + n = 2.100777309 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.642857143' to each side of the equation. 3.642857143 + -3.642857143 + n = 2.100777309 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = 2.100777309 + -3.642857143 n = 2.100777309 + -3.642857143 Combine like terms: 2.100777309 + -3.642857143 = -1.542079834 n = -1.542079834 Simplifying n = -1.542079834Subproblem 2
n + 3.642857143 = -2.100777309 Simplifying n + 3.642857143 = -2.100777309 Reorder the terms: 3.642857143 + n = -2.100777309 Solving 3.642857143 + n = -2.100777309 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.642857143' to each side of the equation. 3.642857143 + -3.642857143 + n = -2.100777309 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = -2.100777309 + -3.642857143 n = -2.100777309 + -3.642857143 Combine like terms: -2.100777309 + -3.642857143 = -5.743634452 n = -5.743634452 Simplifying n = -5.743634452Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.542079834, -5.743634452}
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