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Simplifying 2n(10 + n) = 75 (10 * 2n + n * 2n) = 75 (20n + 2n2) = 75 Solving 20n + 2n2 = 75 Solving for variable 'n'. Reorder the terms: -75 + 20n + 2n2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 20n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -37.5 + 10n + n2 = 0 Move the constant term to the right: Add '37.5' to each side of the equation. -37.5 + 10n + 37.5 + n2 = 0 + 37.5 Reorder the terms: -37.5 + 37.5 + 10n + n2 = 0 + 37.5 Combine like terms: -37.5 + 37.5 = 0.0 0.0 + 10n + n2 = 0 + 37.5 10n + n2 = 0 + 37.5 Combine like terms: 0 + 37.5 = 37.5 10n + n2 = 37.5 The n term is 10n. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10n + 25 + n2 = 37.5 + 25 Reorder the terms: 25 + 10n + n2 = 37.5 + 25 Combine like terms: 37.5 + 25 = 62.5 25 + 10n + n2 = 62.5 Factor a perfect square on the left side: (n + 5)(n + 5) = 62.5 Calculate the square root of the right side: 7.90569415 Break this problem into two subproblems by setting (n + 5) equal to 7.90569415 and -7.90569415.Subproblem 1
n + 5 = 7.90569415 Simplifying n + 5 = 7.90569415 Reorder the terms: 5 + n = 7.90569415 Solving 5 + n = 7.90569415 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 7.90569415 + -5 Combine like terms: 5 + -5 = 0 0 + n = 7.90569415 + -5 n = 7.90569415 + -5 Combine like terms: 7.90569415 + -5 = 2.90569415 n = 2.90569415 Simplifying n = 2.90569415Subproblem 2
n + 5 = -7.90569415 Simplifying n + 5 = -7.90569415 Reorder the terms: 5 + n = -7.90569415 Solving 5 + n = -7.90569415 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -7.90569415 + -5 Combine like terms: 5 + -5 = 0 0 + n = -7.90569415 + -5 n = -7.90569415 + -5 Combine like terms: -7.90569415 + -5 = -12.90569415 n = -12.90569415 Simplifying n = -12.90569415Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.90569415, -12.90569415}
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