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Simplifying 5n + n2 + -420 = 0 Reorder the terms: -420 + 5n + n2 = 0 Solving -420 + 5n + 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 + 5n + 420 + n2 = 0 + 420 Reorder the terms: -420 + 420 + 5n + n2 = 0 + 420 Combine like terms: -420 + 420 = 0 0 + 5n + n2 = 0 + 420 5n + n2 = 0 + 420 Combine like terms: 0 + 420 = 420 5n + n2 = 420 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 420 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 420 + 6.25 Combine like terms: 420 + 6.25 = 426.25 6.25 + 5n + n2 = 426.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 426.25 Calculate the square root of the right side: 20.645822822 Break this problem into two subproblems by setting (n + 2.5) equal to 20.645822822 and -20.645822822.Subproblem 1
n + 2.5 = 20.645822822 Simplifying n + 2.5 = 20.645822822 Reorder the terms: 2.5 + n = 20.645822822 Solving 2.5 + n = 20.645822822 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 20.645822822 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 20.645822822 + -2.5 n = 20.645822822 + -2.5 Combine like terms: 20.645822822 + -2.5 = 18.145822822 n = 18.145822822 Simplifying n = 18.145822822Subproblem 2
n + 2.5 = -20.645822822 Simplifying n + 2.5 = -20.645822822 Reorder the terms: 2.5 + n = -20.645822822 Solving 2.5 + n = -20.645822822 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -20.645822822 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -20.645822822 + -2.5 n = -20.645822822 + -2.5 Combine like terms: -20.645822822 + -2.5 = -23.145822822 n = -23.145822822 Simplifying n = -23.145822822Solution
The solution to the problem is based on the solutions from the subproblems. n = {18.145822822, -23.145822822}
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