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Simplifying n2 + -10n + 23 = 0 Reorder the terms: 23 + -10n + n2 = 0 Solving 23 + -10n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-23' to each side of the equation. 23 + -10n + -23 + n2 = 0 + -23 Reorder the terms: 23 + -23 + -10n + n2 = 0 + -23 Combine like terms: 23 + -23 = 0 0 + -10n + n2 = 0 + -23 -10n + n2 = 0 + -23 Combine like terms: 0 + -23 = -23 -10n + n2 = -23 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 = -23 + 25 Reorder the terms: 25 + -10n + n2 = -23 + 25 Combine like terms: -23 + 25 = 2 25 + -10n + n2 = 2 Factor a perfect square on the left side: (n + -5)(n + -5) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (n + -5) equal to 1.414213562 and -1.414213562.Subproblem 1
n + -5 = 1.414213562 Simplifying n + -5 = 1.414213562 Reorder the terms: -5 + n = 1.414213562 Solving -5 + n = 1.414213562 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 = 1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + n = 1.414213562 + 5 n = 1.414213562 + 5 Combine like terms: 1.414213562 + 5 = 6.414213562 n = 6.414213562 Simplifying n = 6.414213562Subproblem 2
n + -5 = -1.414213562 Simplifying n + -5 = -1.414213562 Reorder the terms: -5 + n = -1.414213562 Solving -5 + n = -1.414213562 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 = -1.414213562 + 5 Combine like terms: -5 + 5 = 0 0 + n = -1.414213562 + 5 n = -1.414213562 + 5 Combine like terms: -1.414213562 + 5 = 3.585786438 n = 3.585786438 Simplifying n = 3.585786438Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.414213562, 3.585786438}
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