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Simplifying 8n2 + -3n + -16 = 24 Reorder the terms: -16 + -3n + 8n2 = 24 Solving -16 + -3n + 8n2 = 24 Solving for variable 'n'. Reorder the terms: -16 + -24 + -3n + 8n2 = 24 + -24 Combine like terms: -16 + -24 = -40 -40 + -3n + 8n2 = 24 + -24 Combine like terms: 24 + -24 = 0 -40 + -3n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -5 + -0.375n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.375n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.375n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.375n + n2 = 0 + 5 -0.375n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.375n + n2 = 5 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 = 5 + 0.03515625 Reorder the terms: 0.03515625 + -0.375n + n2 = 5 + 0.03515625 Combine like terms: 5 + 0.03515625 = 5.03515625 0.03515625 + -0.375n + n2 = 5.03515625 Factor a perfect square on the left side: (n + -0.1875)(n + -0.1875) = 5.03515625 Calculate the square root of the right side: 2.243915384 Break this problem into two subproblems by setting (n + -0.1875) equal to 2.243915384 and -2.243915384.Subproblem 1
n + -0.1875 = 2.243915384 Simplifying n + -0.1875 = 2.243915384 Reorder the terms: -0.1875 + n = 2.243915384 Solving -0.1875 + n = 2.243915384 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 = 2.243915384 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 2.243915384 + 0.1875 n = 2.243915384 + 0.1875 Combine like terms: 2.243915384 + 0.1875 = 2.431415384 n = 2.431415384 Simplifying n = 2.431415384Subproblem 2
n + -0.1875 = -2.243915384 Simplifying n + -0.1875 = -2.243915384 Reorder the terms: -0.1875 + n = -2.243915384 Solving -0.1875 + n = -2.243915384 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 = -2.243915384 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -2.243915384 + 0.1875 n = -2.243915384 + 0.1875 Combine like terms: -2.243915384 + 0.1875 = -2.056415384 n = -2.056415384 Simplifying n = -2.056415384Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.431415384, -2.056415384}
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