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Simplifying -8n(n + -4) = n + -40 Reorder the terms: -8n(-4 + n) = n + -40 (-4 * -8n + n * -8n) = n + -40 (32n + -8n2) = n + -40 Reorder the terms: 32n + -8n2 = -40 + n Solving 32n + -8n2 = -40 + n Solving for variable 'n'. Reorder the terms: 40 + 32n + -1n + -8n2 = -40 + n + 40 + -1n Combine like terms: 32n + -1n = 31n 40 + 31n + -8n2 = -40 + n + 40 + -1n Reorder the terms: 40 + 31n + -8n2 = -40 + 40 + n + -1n Combine like terms: -40 + 40 = 0 40 + 31n + -8n2 = 0 + n + -1n 40 + 31n + -8n2 = n + -1n Combine like terms: n + -1n = 0 40 + 31n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -5 + -3.875n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -3.875n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -3.875n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3.875n + n2 = 0 + 5 -3.875n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3.875n + n2 = 5 The n term is -3.875n. Take half its coefficient (-1.9375). Square it (3.75390625) and add it to both sides. Add '3.75390625' to each side of the equation. -3.875n + 3.75390625 + n2 = 5 + 3.75390625 Reorder the terms: 3.75390625 + -3.875n + n2 = 5 + 3.75390625 Combine like terms: 5 + 3.75390625 = 8.75390625 3.75390625 + -3.875n + n2 = 8.75390625 Factor a perfect square on the left side: (n + -1.9375)(n + -1.9375) = 8.75390625 Calculate the square root of the right side: 2.958700095 Break this problem into two subproblems by setting (n + -1.9375) equal to 2.958700095 and -2.958700095.Subproblem 1
n + -1.9375 = 2.958700095 Simplifying n + -1.9375 = 2.958700095 Reorder the terms: -1.9375 + n = 2.958700095 Solving -1.9375 + n = 2.958700095 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.9375' to each side of the equation. -1.9375 + 1.9375 + n = 2.958700095 + 1.9375 Combine like terms: -1.9375 + 1.9375 = 0.0000 0.0000 + n = 2.958700095 + 1.9375 n = 2.958700095 + 1.9375 Combine like terms: 2.958700095 + 1.9375 = 4.896200095 n = 4.896200095 Simplifying n = 4.896200095Subproblem 2
n + -1.9375 = -2.958700095 Simplifying n + -1.9375 = -2.958700095 Reorder the terms: -1.9375 + n = -2.958700095 Solving -1.9375 + n = -2.958700095 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.9375' to each side of the equation. -1.9375 + 1.9375 + n = -2.958700095 + 1.9375 Combine like terms: -1.9375 + 1.9375 = 0.0000 0.0000 + n = -2.958700095 + 1.9375 n = -2.958700095 + 1.9375 Combine like terms: -2.958700095 + 1.9375 = -1.021200095 n = -1.021200095 Simplifying n = -1.021200095Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.896200095, -1.021200095}
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