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Simplifying 0.25n2 + 13.75n + -500 = 0 Reorder the terms: -500 + 13.75n + 0.25n2 = 0 Solving -500 + 13.75n + 0.25n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.25 the coefficient of the squared term: Divide each side by '0.25'. -2000 + 55n + n2 = 0 Move the constant term to the right: Add '2000' to each side of the equation. -2000 + 55n + 2000 + n2 = 0 + 2000 Reorder the terms: -2000 + 2000 + 55n + n2 = 0 + 2000 Combine like terms: -2000 + 2000 = 0 0 + 55n + n2 = 0 + 2000 55n + n2 = 0 + 2000 Combine like terms: 0 + 2000 = 2000 55n + n2 = 2000 The n term is 55n. Take half its coefficient (27.5). Square it (756.25) and add it to both sides. Add '756.25' to each side of the equation. 55n + 756.25 + n2 = 2000 + 756.25 Reorder the terms: 756.25 + 55n + n2 = 2000 + 756.25 Combine like terms: 2000 + 756.25 = 2756.25 756.25 + 55n + n2 = 2756.25 Factor a perfect square on the left side: (n + 27.5)(n + 27.5) = 2756.25 Calculate the square root of the right side: 52.5 Break this problem into two subproblems by setting (n + 27.5) equal to 52.5 and -52.5.Subproblem 1
n + 27.5 = 52.5 Simplifying n + 27.5 = 52.5 Reorder the terms: 27.5 + n = 52.5 Solving 27.5 + n = 52.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + n = 52.5 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + n = 52.5 + -27.5 n = 52.5 + -27.5 Combine like terms: 52.5 + -27.5 = 25 n = 25 Simplifying n = 25Subproblem 2
n + 27.5 = -52.5 Simplifying n + 27.5 = -52.5 Reorder the terms: 27.5 + n = -52.5 Solving 27.5 + n = -52.5 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + n = -52.5 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + n = -52.5 + -27.5 n = -52.5 + -27.5 Combine like terms: -52.5 + -27.5 = -80 n = -80 Simplifying n = -80Solution
The solution to the problem is based on the solutions from the subproblems. n = {25, -80}
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