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Simplifying n2 + -3n + -420 = 0 Reorder the terms: -420 + -3n + n2 = 0 Solving -420 + -3n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '420' to each side of the equation. -420 + -3n + 420 + n2 = 0 + 420 Reorder the terms: -420 + 420 + -3n + n2 = 0 + 420 Combine like terms: -420 + 420 = 0 0 + -3n + n2 = 0 + 420 -3n + n2 = 0 + 420 Combine like terms: 0 + 420 = 420 -3n + n2 = 420 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 = 420 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 420 + 2.25 Combine like terms: 420 + 2.25 = 422.25 2.25 + -3n + n2 = 422.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 422.25 Calculate the square root of the right side: 20.548722588 Break this problem into two subproblems by setting (n + -1.5) equal to 20.548722588 and -20.548722588.Subproblem 1
n + -1.5 = 20.548722588 Simplifying n + -1.5 = 20.548722588 Reorder the terms: -1.5 + n = 20.548722588 Solving -1.5 + n = 20.548722588 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 = 20.548722588 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 20.548722588 + 1.5 n = 20.548722588 + 1.5 Combine like terms: 20.548722588 + 1.5 = 22.048722588 n = 22.048722588 Simplifying n = 22.048722588Subproblem 2
n + -1.5 = -20.548722588 Simplifying n + -1.5 = -20.548722588 Reorder the terms: -1.5 + n = -20.548722588 Solving -1.5 + n = -20.548722588 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 = -20.548722588 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -20.548722588 + 1.5 n = -20.548722588 + 1.5 Combine like terms: -20.548722588 + 1.5 = -19.048722588 n = -19.048722588 Simplifying n = -19.048722588Solution
The solution to the problem is based on the solutions from the subproblems. n = {22.048722588, -19.048722588}
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