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Simplifying 8(4 + 2n) * n = 7 Reorder the terms for easier multiplication: 8n(4 + 2n) = 7 (4 * 8n + 2n * 8n) = 7 (32n + 16n2) = 7 Solving 32n + 16n2 = 7 Solving for variable 'n'. Reorder the terms: -7 + 32n + 16n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 32n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.4375 + 2n + n2 = 0 Move the constant term to the right: Add '0.4375' to each side of the equation. -0.4375 + 2n + 0.4375 + n2 = 0 + 0.4375 Reorder the terms: -0.4375 + 0.4375 + 2n + n2 = 0 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + 2n + n2 = 0 + 0.4375 2n + n2 = 0 + 0.4375 Combine like terms: 0 + 0.4375 = 0.4375 2n + n2 = 0.4375 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 0.4375 + 1 Reorder the terms: 1 + 2n + n2 = 0.4375 + 1 Combine like terms: 0.4375 + 1 = 1.4375 1 + 2n + n2 = 1.4375 Factor a perfect square on the left side: (n + 1)(n + 1) = 1.4375 Calculate the square root of the right side: 1.198957881 Break this problem into two subproblems by setting (n + 1) equal to 1.198957881 and -1.198957881.Subproblem 1
n + 1 = 1.198957881 Simplifying n + 1 = 1.198957881 Reorder the terms: 1 + n = 1.198957881 Solving 1 + n = 1.198957881 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 1.198957881 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.198957881 + -1 n = 1.198957881 + -1 Combine like terms: 1.198957881 + -1 = 0.198957881 n = 0.198957881 Simplifying n = 0.198957881Subproblem 2
n + 1 = -1.198957881 Simplifying n + 1 = -1.198957881 Reorder the terms: 1 + n = -1.198957881 Solving 1 + n = -1.198957881 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -1.198957881 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.198957881 + -1 n = -1.198957881 + -1 Combine like terms: -1.198957881 + -1 = -2.198957881 n = -2.198957881 Simplifying n = -2.198957881Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.198957881, -2.198957881}
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