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Simplifying 8n2 + 5n + -11 = 0 Reorder the terms: -11 + 5n + 8n2 = 0 Solving -11 + 5n + 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'. -1.375 + 0.625n + n2 = 0 Move the constant term to the right: Add '1.375' to each side of the equation. -1.375 + 0.625n + 1.375 + n2 = 0 + 1.375 Reorder the terms: -1.375 + 1.375 + 0.625n + n2 = 0 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + 0.625n + n2 = 0 + 1.375 0.625n + n2 = 0 + 1.375 Combine like terms: 0 + 1.375 = 1.375 0.625n + n2 = 1.375 The n term is 0.625n. Take half its coefficient (0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. 0.625n + 0.09765625 + n2 = 1.375 + 0.09765625 Reorder the terms: 0.09765625 + 0.625n + n2 = 1.375 + 0.09765625 Combine like terms: 1.375 + 0.09765625 = 1.47265625 0.09765625 + 0.625n + n2 = 1.47265625 Factor a perfect square on the left side: (n + 0.3125)(n + 0.3125) = 1.47265625 Calculate the square root of the right side: 1.21353049 Break this problem into two subproblems by setting (n + 0.3125) equal to 1.21353049 and -1.21353049.Subproblem 1
n + 0.3125 = 1.21353049 Simplifying n + 0.3125 = 1.21353049 Reorder the terms: 0.3125 + n = 1.21353049 Solving 0.3125 + n = 1.21353049 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + n = 1.21353049 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + n = 1.21353049 + -0.3125 n = 1.21353049 + -0.3125 Combine like terms: 1.21353049 + -0.3125 = 0.90103049 n = 0.90103049 Simplifying n = 0.90103049Subproblem 2
n + 0.3125 = -1.21353049 Simplifying n + 0.3125 = -1.21353049 Reorder the terms: 0.3125 + n = -1.21353049 Solving 0.3125 + n = -1.21353049 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3125' to each side of the equation. 0.3125 + -0.3125 + n = -1.21353049 + -0.3125 Combine like terms: 0.3125 + -0.3125 = 0.0000 0.0000 + n = -1.21353049 + -0.3125 n = -1.21353049 + -0.3125 Combine like terms: -1.21353049 + -0.3125 = -1.52603049 n = -1.52603049 Simplifying n = -1.52603049Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.90103049, -1.52603049}
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