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Simplifying 7n2 + 51n + -14 = 0 Reorder the terms: -14 + 51n + 7n2 = 0 Solving -14 + 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'. -2 + 7.285714286n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 7.285714286n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 7.285714286n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 7.285714286n + n2 = 0 + 2 7.285714286n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 7.285714286n + n2 = 2 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 = 2 + 13.27040816 Reorder the terms: 13.27040816 + 7.285714286n + n2 = 2 + 13.27040816 Combine like terms: 2 + 13.27040816 = 15.27040816 13.27040816 + 7.285714286n + n2 = 15.27040816 Factor a perfect square on the left side: (n + 3.642857143)(n + 3.642857143) = 15.27040816 Calculate the square root of the right side: 3.907736961 Break this problem into two subproblems by setting (n + 3.642857143) equal to 3.907736961 and -3.907736961.Subproblem 1
n + 3.642857143 = 3.907736961 Simplifying n + 3.642857143 = 3.907736961 Reorder the terms: 3.642857143 + n = 3.907736961 Solving 3.642857143 + n = 3.907736961 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 = 3.907736961 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = 3.907736961 + -3.642857143 n = 3.907736961 + -3.642857143 Combine like terms: 3.907736961 + -3.642857143 = 0.264879818 n = 0.264879818 Simplifying n = 0.264879818Subproblem 2
n + 3.642857143 = -3.907736961 Simplifying n + 3.642857143 = -3.907736961 Reorder the terms: 3.642857143 + n = -3.907736961 Solving 3.642857143 + n = -3.907736961 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 = -3.907736961 + -3.642857143 Combine like terms: 3.642857143 + -3.642857143 = 0.000000000 0.000000000 + n = -3.907736961 + -3.642857143 n = -3.907736961 + -3.642857143 Combine like terms: -3.907736961 + -3.642857143 = -7.550594104 n = -7.550594104 Simplifying n = -7.550594104Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.264879818, -7.550594104}
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