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Simplifying n2 = 2(7n + -23) Reorder the terms: n2 = 2(-23 + 7n) n2 = (-23 * 2 + 7n * 2) n2 = (-46 + 14n) Solving n2 = -46 + 14n Solving for variable 'n'. Reorder the terms: 46 + -14n + n2 = -46 + 14n + 46 + -14n Reorder the terms: 46 + -14n + n2 = -46 + 46 + 14n + -14n Combine like terms: -46 + 46 = 0 46 + -14n + n2 = 0 + 14n + -14n 46 + -14n + n2 = 14n + -14n Combine like terms: 14n + -14n = 0 46 + -14n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-46' to each side of the equation. 46 + -14n + -46 + n2 = 0 + -46 Reorder the terms: 46 + -46 + -14n + n2 = 0 + -46 Combine like terms: 46 + -46 = 0 0 + -14n + n2 = 0 + -46 -14n + n2 = 0 + -46 Combine like terms: 0 + -46 = -46 -14n + n2 = -46 The n term is -14n. Take half its coefficient (-7). Square it (49) and add it to both sides. Add '49' to each side of the equation. -14n + 49 + n2 = -46 + 49 Reorder the terms: 49 + -14n + n2 = -46 + 49 Combine like terms: -46 + 49 = 3 49 + -14n + n2 = 3 Factor a perfect square on the left side: (n + -7)(n + -7) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (n + -7) equal to 1.732050808 and -1.732050808.Subproblem 1
n + -7 = 1.732050808 Simplifying n + -7 = 1.732050808 Reorder the terms: -7 + n = 1.732050808 Solving -7 + n = 1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + n = 1.732050808 + 7 Combine like terms: -7 + 7 = 0 0 + n = 1.732050808 + 7 n = 1.732050808 + 7 Combine like terms: 1.732050808 + 7 = 8.732050808 n = 8.732050808 Simplifying n = 8.732050808Subproblem 2
n + -7 = -1.732050808 Simplifying n + -7 = -1.732050808 Reorder the terms: -7 + n = -1.732050808 Solving -7 + n = -1.732050808 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + n = -1.732050808 + 7 Combine like terms: -7 + 7 = 0 0 + n = -1.732050808 + 7 n = -1.732050808 + 7 Combine like terms: -1.732050808 + 7 = 5.267949192 n = 5.267949192 Simplifying n = 5.267949192Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.732050808, 5.267949192}
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