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Simplifying n2 + 3n + -14 = 0 Reorder the terms: -14 + 3n + n2 = 0 Solving -14 + 3n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + 3n + 14 + n2 = 0 + 14 Reorder the terms: -14 + 14 + 3n + n2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + 3n + n2 = 0 + 14 3n + n2 = 0 + 14 Combine like terms: 0 + 14 = 14 3n + n2 = 14 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 = 14 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 14 + 2.25 Combine like terms: 14 + 2.25 = 16.25 2.25 + 3n + n2 = 16.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (n + 1.5) equal to 4.031128874 and -4.031128874.Subproblem 1
n + 1.5 = 4.031128874 Simplifying n + 1.5 = 4.031128874 Reorder the terms: 1.5 + n = 4.031128874 Solving 1.5 + n = 4.031128874 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 = 4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 4.031128874 + -1.5 n = 4.031128874 + -1.5 Combine like terms: 4.031128874 + -1.5 = 2.531128874 n = 2.531128874 Simplifying n = 2.531128874Subproblem 2
n + 1.5 = -4.031128874 Simplifying n + 1.5 = -4.031128874 Reorder the terms: 1.5 + n = -4.031128874 Solving 1.5 + n = -4.031128874 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 = -4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -4.031128874 + -1.5 n = -4.031128874 + -1.5 Combine like terms: -4.031128874 + -1.5 = -5.531128874 n = -5.531128874 Simplifying n = -5.531128874Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.531128874, -5.531128874}
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