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Simplifying 2n(7n + 8) = n + 8 Reorder the terms: 2n(8 + 7n) = n + 8 (8 * 2n + 7n * 2n) = n + 8 (16n + 14n2) = n + 8 Reorder the terms: 16n + 14n2 = 8 + n Solving 16n + 14n2 = 8 + n Solving for variable 'n'. Reorder the terms: -8 + 16n + -1n + 14n2 = 8 + n + -8 + -1n Combine like terms: 16n + -1n = 15n -8 + 15n + 14n2 = 8 + n + -8 + -1n Reorder the terms: -8 + 15n + 14n2 = 8 + -8 + n + -1n Combine like terms: 8 + -8 = 0 -8 + 15n + 14n2 = 0 + n + -1n -8 + 15n + 14n2 = n + -1n Combine like terms: n + -1n = 0 -8 + 15n + 14n2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.5714285714 + 1.071428571n + n2 = 0 Move the constant term to the right: Add '0.5714285714' to each side of the equation. -0.5714285714 + 1.071428571n + 0.5714285714 + n2 = 0 + 0.5714285714 Reorder the terms: -0.5714285714 + 0.5714285714 + 1.071428571n + n2 = 0 + 0.5714285714 Combine like terms: -0.5714285714 + 0.5714285714 = 0.0000000000 0.0000000000 + 1.071428571n + n2 = 0 + 0.5714285714 1.071428571n + n2 = 0 + 0.5714285714 Combine like terms: 0 + 0.5714285714 = 0.5714285714 1.071428571n + n2 = 0.5714285714 The n term is 1.071428571n. Take half its coefficient (0.5357142855). Square it (0.2869897957) and add it to both sides. Add '0.2869897957' to each side of the equation. 1.071428571n + 0.2869897957 + n2 = 0.5714285714 + 0.2869897957 Reorder the terms: 0.2869897957 + 1.071428571n + n2 = 0.5714285714 + 0.2869897957 Combine like terms: 0.5714285714 + 0.2869897957 = 0.8584183671 0.2869897957 + 1.071428571n + n2 = 0.8584183671 Factor a perfect square on the left side: (n + 0.5357142855)(n + 0.5357142855) = 0.8584183671 Calculate the square root of the right side: 0.926508698 Break this problem into two subproblems by setting (n + 0.5357142855) equal to 0.926508698 and -0.926508698.Subproblem 1
n + 0.5357142855 = 0.926508698 Simplifying n + 0.5357142855 = 0.926508698 Reorder the terms: 0.5357142855 + n = 0.926508698 Solving 0.5357142855 + n = 0.926508698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5357142855' to each side of the equation. 0.5357142855 + -0.5357142855 + n = 0.926508698 + -0.5357142855 Combine like terms: 0.5357142855 + -0.5357142855 = 0.0000000000 0.0000000000 + n = 0.926508698 + -0.5357142855 n = 0.926508698 + -0.5357142855 Combine like terms: 0.926508698 + -0.5357142855 = 0.3907944125 n = 0.3907944125 Simplifying n = 0.3907944125Subproblem 2
n + 0.5357142855 = -0.926508698 Simplifying n + 0.5357142855 = -0.926508698 Reorder the terms: 0.5357142855 + n = -0.926508698 Solving 0.5357142855 + n = -0.926508698 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5357142855' to each side of the equation. 0.5357142855 + -0.5357142855 + n = -0.926508698 + -0.5357142855 Combine like terms: 0.5357142855 + -0.5357142855 = 0.0000000000 0.0000000000 + n = -0.926508698 + -0.5357142855 n = -0.926508698 + -0.5357142855 Combine like terms: -0.926508698 + -0.5357142855 = -1.4622229835 n = -1.4622229835 Simplifying n = -1.4622229835Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.3907944125, -1.4622229835}
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