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Simplifying 8n2 + -21n + -24 = 0 Reorder the terms: -24 + -21n + 8n2 = 0 Solving -24 + -21n + 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'. -3 + -2.625n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -2.625n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + -2.625n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -2.625n + n2 = 0 + 3 -2.625n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 -2.625n + n2 = 3 The n term is -2.625n. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625n + 1.72265625 + n2 = 3 + 1.72265625 Reorder the terms: 1.72265625 + -2.625n + n2 = 3 + 1.72265625 Combine like terms: 3 + 1.72265625 = 4.72265625 1.72265625 + -2.625n + n2 = 4.72265625 Factor a perfect square on the left side: (n + -1.3125)(n + -1.3125) = 4.72265625 Calculate the square root of the right side: 2.173167331 Break this problem into two subproblems by setting (n + -1.3125) equal to 2.173167331 and -2.173167331.Subproblem 1
n + -1.3125 = 2.173167331 Simplifying n + -1.3125 = 2.173167331 Reorder the terms: -1.3125 + n = 2.173167331 Solving -1.3125 + n = 2.173167331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + n = 2.173167331 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + n = 2.173167331 + 1.3125 n = 2.173167331 + 1.3125 Combine like terms: 2.173167331 + 1.3125 = 3.485667331 n = 3.485667331 Simplifying n = 3.485667331Subproblem 2
n + -1.3125 = -2.173167331 Simplifying n + -1.3125 = -2.173167331 Reorder the terms: -1.3125 + n = -2.173167331 Solving -1.3125 + n = -2.173167331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + n = -2.173167331 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + n = -2.173167331 + 1.3125 n = -2.173167331 + 1.3125 Combine like terms: -2.173167331 + 1.3125 = -0.860667331 n = -0.860667331 Simplifying n = -0.860667331Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.485667331, -0.860667331}
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