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Simplifying 2n2 + 23n + -420 = 0 Reorder the terms: -420 + 23n + 2n2 = 0 Solving -420 + 23n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -210 + 11.5n + n2 = 0 Move the constant term to the right: Add '210' to each side of the equation. -210 + 11.5n + 210 + n2 = 0 + 210 Reorder the terms: -210 + 210 + 11.5n + n2 = 0 + 210 Combine like terms: -210 + 210 = 0 0 + 11.5n + n2 = 0 + 210 11.5n + n2 = 0 + 210 Combine like terms: 0 + 210 = 210 11.5n + n2 = 210 The n term is 11.5n. Take half its coefficient (5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. 11.5n + 33.0625 + n2 = 210 + 33.0625 Reorder the terms: 33.0625 + 11.5n + n2 = 210 + 33.0625 Combine like terms: 210 + 33.0625 = 243.0625 33.0625 + 11.5n + n2 = 243.0625 Factor a perfect square on the left side: (n + 5.75)(n + 5.75) = 243.0625 Calculate the square root of the right side: 15.590461828 Break this problem into two subproblems by setting (n + 5.75) equal to 15.590461828 and -15.590461828.Subproblem 1
n + 5.75 = 15.590461828 Simplifying n + 5.75 = 15.590461828 Reorder the terms: 5.75 + n = 15.590461828 Solving 5.75 + n = 15.590461828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = 15.590461828 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = 15.590461828 + -5.75 n = 15.590461828 + -5.75 Combine like terms: 15.590461828 + -5.75 = 9.840461828 n = 9.840461828 Simplifying n = 9.840461828Subproblem 2
n + 5.75 = -15.590461828 Simplifying n + 5.75 = -15.590461828 Reorder the terms: 5.75 + n = -15.590461828 Solving 5.75 + n = -15.590461828 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = -15.590461828 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = -15.590461828 + -5.75 n = -15.590461828 + -5.75 Combine like terms: -15.590461828 + -5.75 = -21.340461828 n = -21.340461828 Simplifying n = -21.340461828Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.840461828, -21.340461828}
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