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Simplifying 2n2 + 11n = 75 Reorder the terms: 11n + 2n2 = 75 Solving 11n + 2n2 = 75 Solving for variable 'n'. Reorder the terms: -75 + 11n + 2n2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 11n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -37.5 + 5.5n + n2 = 0 Move the constant term to the right: Add '37.5' to each side of the equation. -37.5 + 5.5n + 37.5 + n2 = 0 + 37.5 Reorder the terms: -37.5 + 37.5 + 5.5n + n2 = 0 + 37.5 Combine like terms: -37.5 + 37.5 = 0.0 0.0 + 5.5n + n2 = 0 + 37.5 5.5n + n2 = 0 + 37.5 Combine like terms: 0 + 37.5 = 37.5 5.5n + n2 = 37.5 The n term is 5.5n. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5n + 7.5625 + n2 = 37.5 + 7.5625 Reorder the terms: 7.5625 + 5.5n + n2 = 37.5 + 7.5625 Combine like terms: 37.5 + 7.5625 = 45.0625 7.5625 + 5.5n + n2 = 45.0625 Factor a perfect square on the left side: (n + 2.75)(n + 2.75) = 45.0625 Calculate the square root of the right side: 6.712860791 Break this problem into two subproblems by setting (n + 2.75) equal to 6.712860791 and -6.712860791.Subproblem 1
n + 2.75 = 6.712860791 Simplifying n + 2.75 = 6.712860791 Reorder the terms: 2.75 + n = 6.712860791 Solving 2.75 + n = 6.712860791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + n = 6.712860791 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = 6.712860791 + -2.75 n = 6.712860791 + -2.75 Combine like terms: 6.712860791 + -2.75 = 3.962860791 n = 3.962860791 Simplifying n = 3.962860791Subproblem 2
n + 2.75 = -6.712860791 Simplifying n + 2.75 = -6.712860791 Reorder the terms: 2.75 + n = -6.712860791 Solving 2.75 + n = -6.712860791 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + n = -6.712860791 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + n = -6.712860791 + -2.75 n = -6.712860791 + -2.75 Combine like terms: -6.712860791 + -2.75 = -9.462860791 n = -9.462860791 Simplifying n = -9.462860791Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.962860791, -9.462860791}
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