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Simplifying 8n(4n + 7) = 17 Reorder the terms: 8n(7 + 4n) = 17 (7 * 8n + 4n * 8n) = 17 (56n + 32n2) = 17 Solving 56n + 32n2 = 17 Solving for variable 'n'. Reorder the terms: -17 + 56n + 32n2 = 17 + -17 Combine like terms: 17 + -17 = 0 -17 + 56n + 32n2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.53125 + 1.75n + n2 = 0 Move the constant term to the right: Add '0.53125' to each side of the equation. -0.53125 + 1.75n + 0.53125 + n2 = 0 + 0.53125 Reorder the terms: -0.53125 + 0.53125 + 1.75n + n2 = 0 + 0.53125 Combine like terms: -0.53125 + 0.53125 = 0.00000 0.00000 + 1.75n + n2 = 0 + 0.53125 1.75n + n2 = 0 + 0.53125 Combine like terms: 0 + 0.53125 = 0.53125 1.75n + n2 = 0.53125 The n term is 1.75n. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75n + 0.765625 + n2 = 0.53125 + 0.765625 Reorder the terms: 0.765625 + 1.75n + n2 = 0.53125 + 0.765625 Combine like terms: 0.53125 + 0.765625 = 1.296875 0.765625 + 1.75n + n2 = 1.296875 Factor a perfect square on the left side: (n + 0.875)(n + 0.875) = 1.296875 Calculate the square root of the right side: 1.138804197 Break this problem into two subproblems by setting (n + 0.875) equal to 1.138804197 and -1.138804197.Subproblem 1
n + 0.875 = 1.138804197 Simplifying n + 0.875 = 1.138804197 Reorder the terms: 0.875 + n = 1.138804197 Solving 0.875 + n = 1.138804197 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 1.138804197 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.138804197 + -0.875 n = 1.138804197 + -0.875 Combine like terms: 1.138804197 + -0.875 = 0.263804197 n = 0.263804197 Simplifying n = 0.263804197Subproblem 2
n + 0.875 = -1.138804197 Simplifying n + 0.875 = -1.138804197 Reorder the terms: 0.875 + n = -1.138804197 Solving 0.875 + n = -1.138804197 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -1.138804197 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.138804197 + -0.875 n = -1.138804197 + -0.875 Combine like terms: -1.138804197 + -0.875 = -2.013804197 n = -2.013804197 Simplifying n = -2.013804197Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.263804197, -2.013804197}
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