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Simplifying 8n2 + -3n = 10 Reorder the terms: -3n + 8n2 = 10 Solving -3n + 8n2 = 10 Solving for variable 'n'. Reorder the terms: -10 + -3n + 8n2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + -3n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.25 + -0.375n + n2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -0.375n + 1.25 + n2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -0.375n + n2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -0.375n + n2 = 0 + 1.25 -0.375n + n2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -0.375n + n2 = 1.25 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 = 1.25 + 0.03515625 Reorder the terms: 0.03515625 + -0.375n + n2 = 1.25 + 0.03515625 Combine like terms: 1.25 + 0.03515625 = 1.28515625 0.03515625 + -0.375n + n2 = 1.28515625 Factor a perfect square on the left side: (n + -0.1875)(n + -0.1875) = 1.28515625 Calculate the square root of the right side: 1.133647322 Break this problem into two subproblems by setting (n + -0.1875) equal to 1.133647322 and -1.133647322.Subproblem 1
n + -0.1875 = 1.133647322 Simplifying n + -0.1875 = 1.133647322 Reorder the terms: -0.1875 + n = 1.133647322 Solving -0.1875 + n = 1.133647322 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 = 1.133647322 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 1.133647322 + 0.1875 n = 1.133647322 + 0.1875 Combine like terms: 1.133647322 + 0.1875 = 1.321147322 n = 1.321147322 Simplifying n = 1.321147322Subproblem 2
n + -0.1875 = -1.133647322 Simplifying n + -0.1875 = -1.133647322 Reorder the terms: -0.1875 + n = -1.133647322 Solving -0.1875 + n = -1.133647322 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 = -1.133647322 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -1.133647322 + 0.1875 n = -1.133647322 + 0.1875 Combine like terms: -1.133647322 + 0.1875 = -0.946147322 n = -0.946147322 Simplifying n = -0.946147322Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.321147322, -0.946147322}
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