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Simplifying n2 + 7n + 5 = 0 Reorder the terms: 5 + 7n + n2 = 0 Solving 5 + 7n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 7n + -5 + n2 = 0 + -5 Reorder the terms: 5 + -5 + 7n + n2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 7n + n2 = 0 + -5 7n + n2 = 0 + -5 Combine like terms: 0 + -5 = -5 7n + n2 = -5 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = -5 + 12.25 Reorder the terms: 12.25 + 7n + n2 = -5 + 12.25 Combine like terms: -5 + 12.25 = 7.25 12.25 + 7n + n2 = 7.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (n + 3.5) equal to 2.692582404 and -2.692582404.Subproblem 1
n + 3.5 = 2.692582404 Simplifying n + 3.5 = 2.692582404 Reorder the terms: 3.5 + n = 2.692582404 Solving 3.5 + n = 2.692582404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 2.692582404 + -3.5 n = 2.692582404 + -3.5 Combine like terms: 2.692582404 + -3.5 = -0.807417596 n = -0.807417596 Simplifying n = -0.807417596Subproblem 2
n + 3.5 = -2.692582404 Simplifying n + 3.5 = -2.692582404 Reorder the terms: 3.5 + n = -2.692582404 Solving 3.5 + n = -2.692582404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -2.692582404 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -2.692582404 + -3.5 n = -2.692582404 + -3.5 Combine like terms: -2.692582404 + -3.5 = -6.192582404 n = -6.192582404 Simplifying n = -6.192582404Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.807417596, -6.192582404}
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