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Simplifying n(2n + 3) = 75 Reorder the terms: n(3 + 2n) = 75 (3 * n + 2n * n) = 75 (3n + 2n2) = 75 Solving 3n + 2n2 = 75 Solving for variable 'n'. Reorder the terms: -75 + 3n + 2n2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -37.5 + 1.5n + n2 = 0 Move the constant term to the right: Add '37.5' to each side of the equation. -37.5 + 1.5n + 37.5 + n2 = 0 + 37.5 Reorder the terms: -37.5 + 37.5 + 1.5n + n2 = 0 + 37.5 Combine like terms: -37.5 + 37.5 = 0.0 0.0 + 1.5n + n2 = 0 + 37.5 1.5n + n2 = 0 + 37.5 Combine like terms: 0 + 37.5 = 37.5 1.5n + n2 = 37.5 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = 37.5 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 37.5 + 0.5625 Combine like terms: 37.5 + 0.5625 = 38.0625 0.5625 + 1.5n + n2 = 38.0625 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 38.0625 Calculate the square root of the right side: 6.16948134 Break this problem into two subproblems by setting (n + 0.75) equal to 6.16948134 and -6.16948134.Subproblem 1
n + 0.75 = 6.16948134 Simplifying n + 0.75 = 6.16948134 Reorder the terms: 0.75 + n = 6.16948134 Solving 0.75 + n = 6.16948134 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 6.16948134 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 6.16948134 + -0.75 n = 6.16948134 + -0.75 Combine like terms: 6.16948134 + -0.75 = 5.41948134 n = 5.41948134 Simplifying n = 5.41948134Subproblem 2
n + 0.75 = -6.16948134 Simplifying n + 0.75 = -6.16948134 Reorder the terms: 0.75 + n = -6.16948134 Solving 0.75 + n = -6.16948134 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -6.16948134 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -6.16948134 + -0.75 n = -6.16948134 + -0.75 Combine like terms: -6.16948134 + -0.75 = -6.91948134 n = -6.91948134 Simplifying n = -6.91948134Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.41948134, -6.91948134}
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