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Simplifying 5(7n + 32) + 8(n + 1) * n = 425 Reorder the terms: 5(32 + 7n) + 8(n + 1) * n = 425 (32 * 5 + 7n * 5) + 8(n + 1) * n = 425 (160 + 35n) + 8(n + 1) * n = 425 Reorder the terms: 160 + 35n + 8(1 + n) * n = 425 Reorder the terms for easier multiplication: 160 + 35n + 8n(1 + n) = 425 160 + 35n + (1 * 8n + n * 8n) = 425 160 + 35n + (8n + 8n2) = 425 Combine like terms: 35n + 8n = 43n 160 + 43n + 8n2 = 425 Solving 160 + 43n + 8n2 = 425 Solving for variable 'n'. Reorder the terms: 160 + -425 + 43n + 8n2 = 425 + -425 Combine like terms: 160 + -425 = -265 -265 + 43n + 8n2 = 425 + -425 Combine like terms: 425 + -425 = 0 -265 + 43n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -33.125 + 5.375n + n2 = 0 Move the constant term to the right: Add '33.125' to each side of the equation. -33.125 + 5.375n + 33.125 + n2 = 0 + 33.125 Reorder the terms: -33.125 + 33.125 + 5.375n + n2 = 0 + 33.125 Combine like terms: -33.125 + 33.125 = 0.000 0.000 + 5.375n + n2 = 0 + 33.125 5.375n + n2 = 0 + 33.125 Combine like terms: 0 + 33.125 = 33.125 5.375n + n2 = 33.125 The n term is 5.375n. Take half its coefficient (2.6875). Square it (7.22265625) and add it to both sides. Add '7.22265625' to each side of the equation. 5.375n + 7.22265625 + n2 = 33.125 + 7.22265625 Reorder the terms: 7.22265625 + 5.375n + n2 = 33.125 + 7.22265625 Combine like terms: 33.125 + 7.22265625 = 40.34765625 7.22265625 + 5.375n + n2 = 40.34765625 Factor a perfect square on the left side: (n + 2.6875)(n + 2.6875) = 40.34765625 Calculate the square root of the right side: 6.351980498 Break this problem into two subproblems by setting (n + 2.6875) equal to 6.351980498 and -6.351980498.Subproblem 1
n + 2.6875 = 6.351980498 Simplifying n + 2.6875 = 6.351980498 Reorder the terms: 2.6875 + n = 6.351980498 Solving 2.6875 + n = 6.351980498 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.6875' to each side of the equation. 2.6875 + -2.6875 + n = 6.351980498 + -2.6875 Combine like terms: 2.6875 + -2.6875 = 0.0000 0.0000 + n = 6.351980498 + -2.6875 n = 6.351980498 + -2.6875 Combine like terms: 6.351980498 + -2.6875 = 3.664480498 n = 3.664480498 Simplifying n = 3.664480498Subproblem 2
n + 2.6875 = -6.351980498 Simplifying n + 2.6875 = -6.351980498 Reorder the terms: 2.6875 + n = -6.351980498 Solving 2.6875 + n = -6.351980498 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.6875' to each side of the equation. 2.6875 + -2.6875 + n = -6.351980498 + -2.6875 Combine like terms: 2.6875 + -2.6875 = 0.0000 0.0000 + n = -6.351980498 + -2.6875 n = -6.351980498 + -2.6875 Combine like terms: -6.351980498 + -2.6875 = -9.039480498 n = -9.039480498 Simplifying n = -9.039480498Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.664480498, -9.039480498}
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