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Simplifying (5 + n)(15 + n)(0.5) = 0 Reorder the terms for easier multiplication: 0.5(5 + n)(15 + n) = 0 Multiply (5 + n) * (15 + n) 0.5(5(15 + n) + n(15 + n)) = 0 0.5((15 * 5 + n * 5) + n(15 + n)) = 0 0.5((75 + 5n) + n(15 + n)) = 0 0.5(75 + 5n + (15 * n + n * n)) = 0 0.5(75 + 5n + (15n + n2)) = 0 Combine like terms: 5n + 15n = 20n 0.5(75 + 20n + n2) = 0 (75 * 0.5 + 20n * 0.5 + n2 * 0.5) = 0 (37.5 + 10n + 0.5n2) = 0 Solving 37.5 + 10n + 0.5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 75 + 20n + n2 = 0 Move the constant term to the right: Add '-75' to each side of the equation. 75 + 20n + -75 + n2 = 0 + -75 Reorder the terms: 75 + -75 + 20n + n2 = 0 + -75 Combine like terms: 75 + -75 = 0 0 + 20n + n2 = 0 + -75 20n + n2 = 0 + -75 Combine like terms: 0 + -75 = -75 20n + n2 = -75 The n term is 20n. Take half its coefficient (10). Square it (100) and add it to both sides. Add '100' to each side of the equation. 20n + 100 + n2 = -75 + 100 Reorder the terms: 100 + 20n + n2 = -75 + 100 Combine like terms: -75 + 100 = 25 100 + 20n + n2 = 25 Factor a perfect square on the left side: (n + 10)(n + 10) = 25 Calculate the square root of the right side: 5 Break this problem into two subproblems by setting (n + 10) equal to 5 and -5.Subproblem 1
n + 10 = 5 Simplifying n + 10 = 5 Reorder the terms: 10 + n = 5 Solving 10 + n = 5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + n = 5 + -10 Combine like terms: 10 + -10 = 0 0 + n = 5 + -10 n = 5 + -10 Combine like terms: 5 + -10 = -5 n = -5 Simplifying n = -5Subproblem 2
n + 10 = -5 Simplifying n + 10 = -5 Reorder the terms: 10 + n = -5 Solving 10 + n = -5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + n = -5 + -10 Combine like terms: 10 + -10 = 0 0 + n = -5 + -10 n = -5 + -10 Combine like terms: -5 + -10 = -15 n = -15 Simplifying n = -15Solution
The solution to the problem is based on the solutions from the subproblems. n = {-5, -15}
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