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Simplifying n(n + 3) = 323 Reorder the terms: n(3 + n) = 323 (3 * n + n * n) = 323 (3n + n2) = 323 Solving 3n + n2 = 323 Solving for variable 'n'. Reorder the terms: -323 + 3n + n2 = 323 + -323 Combine like terms: 323 + -323 = 0 -323 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '323' to each side of the equation. -323 + 3n + 323 + n2 = 0 + 323 Reorder the terms: -323 + 323 + 3n + n2 = 0 + 323 Combine like terms: -323 + 323 = 0 0 + 3n + n2 = 0 + 323 3n + n2 = 0 + 323 Combine like terms: 0 + 323 = 323 3n + n2 = 323 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 323 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 323 + 2.25 Combine like terms: 323 + 2.25 = 325.25 2.25 + 3n + n2 = 325.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 325.25 Calculate the square root of the right side: 18.034688797 Break this problem into two subproblems by setting (n + 1.5) equal to 18.034688797 and -18.034688797.Subproblem 1
n + 1.5 = 18.034688797 Simplifying n + 1.5 = 18.034688797 Reorder the terms: 1.5 + n = 18.034688797 Solving 1.5 + n = 18.034688797 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 18.034688797 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 18.034688797 + -1.5 n = 18.034688797 + -1.5 Combine like terms: 18.034688797 + -1.5 = 16.534688797 n = 16.534688797 Simplifying n = 16.534688797Subproblem 2
n + 1.5 = -18.034688797 Simplifying n + 1.5 = -18.034688797 Reorder the terms: 1.5 + n = -18.034688797 Solving 1.5 + n = -18.034688797 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -18.034688797 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -18.034688797 + -1.5 n = -18.034688797 + -1.5 Combine like terms: -18.034688797 + -1.5 = -19.534688797 n = -19.534688797 Simplifying n = -19.534688797Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.534688797, -19.534688797}
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