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Simplifying n2 + 3n + -80 = 0 Reorder the terms: -80 + 3n + n2 = 0 Solving -80 + 3n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '80' to each side of the equation. -80 + 3n + 80 + n2 = 0 + 80 Reorder the terms: -80 + 80 + 3n + n2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + 3n + n2 = 0 + 80 3n + n2 = 0 + 80 Combine like terms: 0 + 80 = 80 3n + n2 = 80 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 = 80 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 80 + 2.25 Combine like terms: 80 + 2.25 = 82.25 2.25 + 3n + n2 = 82.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 82.25 Calculate the square root of the right side: 9.069178574 Break this problem into two subproblems by setting (n + 1.5) equal to 9.069178574 and -9.069178574.Subproblem 1
n + 1.5 = 9.069178574 Simplifying n + 1.5 = 9.069178574 Reorder the terms: 1.5 + n = 9.069178574 Solving 1.5 + n = 9.069178574 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 = 9.069178574 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 9.069178574 + -1.5 n = 9.069178574 + -1.5 Combine like terms: 9.069178574 + -1.5 = 7.569178574 n = 7.569178574 Simplifying n = 7.569178574Subproblem 2
n + 1.5 = -9.069178574 Simplifying n + 1.5 = -9.069178574 Reorder the terms: 1.5 + n = -9.069178574 Solving 1.5 + n = -9.069178574 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 = -9.069178574 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -9.069178574 + -1.5 n = -9.069178574 + -1.5 Combine like terms: -9.069178574 + -1.5 = -10.569178574 n = -10.569178574 Simplifying n = -10.569178574Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.569178574, -10.569178574}
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