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Simplifying 8n2 + -80n + 193 = 0 Reorder the terms: 193 + -80n + 8n2 = 0 Solving 193 + -80n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 24.125 + -10n + n2 = 0 Move the constant term to the right: Add '-24.125' to each side of the equation. 24.125 + -10n + -24.125 + n2 = 0 + -24.125 Reorder the terms: 24.125 + -24.125 + -10n + n2 = 0 + -24.125 Combine like terms: 24.125 + -24.125 = 0.000 0.000 + -10n + n2 = 0 + -24.125 -10n + n2 = 0 + -24.125 Combine like terms: 0 + -24.125 = -24.125 -10n + n2 = -24.125 The n term is -10n. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10n + 25 + n2 = -24.125 + 25 Reorder the terms: 25 + -10n + n2 = -24.125 + 25 Combine like terms: -24.125 + 25 = 0.875 25 + -10n + n2 = 0.875 Factor a perfect square on the left side: (n + -5)(n + -5) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (n + -5) equal to 0.935414347 and -0.935414347.Subproblem 1
n + -5 = 0.935414347 Simplifying n + -5 = 0.935414347 Reorder the terms: -5 + n = 0.935414347 Solving -5 + n = 0.935414347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = 0.935414347 + 5 Combine like terms: -5 + 5 = 0 0 + n = 0.935414347 + 5 n = 0.935414347 + 5 Combine like terms: 0.935414347 + 5 = 5.935414347 n = 5.935414347 Simplifying n = 5.935414347Subproblem 2
n + -5 = -0.935414347 Simplifying n + -5 = -0.935414347 Reorder the terms: -5 + n = -0.935414347 Solving -5 + n = -0.935414347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = -0.935414347 + 5 Combine like terms: -5 + 5 = 0 0 + n = -0.935414347 + 5 n = -0.935414347 + 5 Combine like terms: -0.935414347 + 5 = 4.064585653 n = 4.064585653 Simplifying n = 4.064585653Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.935414347, 4.064585653}
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