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Simplifying 8n2 + -3n + -8 = 0 Reorder the terms: -8 + -3n + 8n2 = 0 Solving -8 + -3n + 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 + -0.375n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.375n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.375n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.375n + n2 = 0 + 1 -0.375n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.375n + n2 = 1 The n term is -0.375n. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375n + 0.03515625 + n2 = 1 + 0.03515625 Reorder the terms: 0.03515625 + -0.375n + n2 = 1 + 0.03515625 Combine like terms: 1 + 0.03515625 = 1.03515625 0.03515625 + -0.375n + n2 = 1.03515625 Factor a perfect square on the left side: (n + -0.1875)(n + -0.1875) = 1.03515625 Calculate the square root of the right side: 1.017426287 Break this problem into two subproblems by setting (n + -0.1875) equal to 1.017426287 and -1.017426287.Subproblem 1
n + -0.1875 = 1.017426287 Simplifying n + -0.1875 = 1.017426287 Reorder the terms: -0.1875 + n = 1.017426287 Solving -0.1875 + n = 1.017426287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = 1.017426287 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 1.017426287 + 0.1875 n = 1.017426287 + 0.1875 Combine like terms: 1.017426287 + 0.1875 = 1.204926287 n = 1.204926287 Simplifying n = 1.204926287Subproblem 2
n + -0.1875 = -1.017426287 Simplifying n + -0.1875 = -1.017426287 Reorder the terms: -0.1875 + n = -1.017426287 Solving -0.1875 + n = -1.017426287 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = -1.017426287 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -1.017426287 + 0.1875 n = -1.017426287 + 0.1875 Combine like terms: -1.017426287 + 0.1875 = -0.829926287 n = -0.829926287 Simplifying n = -0.829926287Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.204926287, -0.829926287}
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