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Simplifying 4n2 + -8n + -23 = 0 Reorder the terms: -23 + -8n + 4n2 = 0 Solving -23 + -8n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5.75 + -2n + n2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + -2n + 5.75 + n2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + -2n + n2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + -2n + n2 = 0 + 5.75 -2n + n2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 -2n + n2 = 5.75 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 5.75 + 1 Reorder the terms: 1 + -2n + n2 = 5.75 + 1 Combine like terms: 5.75 + 1 = 6.75 1 + -2n + n2 = 6.75 Factor a perfect square on the left side: (n + -1)(n + -1) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (n + -1) equal to 2.598076211 and -2.598076211.Subproblem 1
n + -1 = 2.598076211 Simplifying n + -1 = 2.598076211 Reorder the terms: -1 + n = 2.598076211 Solving -1 + n = 2.598076211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 2.598076211 + 1 Combine like terms: -1 + 1 = 0 0 + n = 2.598076211 + 1 n = 2.598076211 + 1 Combine like terms: 2.598076211 + 1 = 3.598076211 n = 3.598076211 Simplifying n = 3.598076211Subproblem 2
n + -1 = -2.598076211 Simplifying n + -1 = -2.598076211 Reorder the terms: -1 + n = -2.598076211 Solving -1 + n = -2.598076211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -2.598076211 + 1 Combine like terms: -1 + 1 = 0 0 + n = -2.598076211 + 1 n = -2.598076211 + 1 Combine like terms: -2.598076211 + 1 = -1.598076211 n = -1.598076211 Simplifying n = -1.598076211Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.598076211, -1.598076211}
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