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Simplifying n(n + 3) = 85 Reorder the terms: n(3 + n) = 85 (3 * n + n * n) = 85 (3n + n2) = 85 Solving 3n + n2 = 85 Solving for variable 'n'. Reorder the terms: -85 + 3n + n2 = 85 + -85 Combine like terms: 85 + -85 = 0 -85 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '85' to each side of the equation. -85 + 3n + 85 + n2 = 0 + 85 Reorder the terms: -85 + 85 + 3n + n2 = 0 + 85 Combine like terms: -85 + 85 = 0 0 + 3n + n2 = 0 + 85 3n + n2 = 0 + 85 Combine like terms: 0 + 85 = 85 3n + n2 = 85 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 = 85 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 85 + 2.25 Combine like terms: 85 + 2.25 = 87.25 2.25 + 3n + n2 = 87.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 87.25 Calculate the square root of the right side: 9.340770846 Break this problem into two subproblems by setting (n + 1.5) equal to 9.340770846 and -9.340770846.Subproblem 1
n + 1.5 = 9.340770846 Simplifying n + 1.5 = 9.340770846 Reorder the terms: 1.5 + n = 9.340770846 Solving 1.5 + n = 9.340770846 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 = 9.340770846 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 9.340770846 + -1.5 n = 9.340770846 + -1.5 Combine like terms: 9.340770846 + -1.5 = 7.840770846 n = 7.840770846 Simplifying n = 7.840770846Subproblem 2
n + 1.5 = -9.340770846 Simplifying n + 1.5 = -9.340770846 Reorder the terms: 1.5 + n = -9.340770846 Solving 1.5 + n = -9.340770846 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 = -9.340770846 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -9.340770846 + -1.5 n = -9.340770846 + -1.5 Combine like terms: -9.340770846 + -1.5 = -10.840770846 n = -10.840770846 Simplifying n = -10.840770846Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.840770846, -10.840770846}
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