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Simplifying 4n2 + -3n + -8 = 12 Reorder the terms: -8 + -3n + 4n2 = 12 Solving -8 + -3n + 4n2 = 12 Solving for variable 'n'. Reorder the terms: -8 + -12 + -3n + 4n2 = 12 + -12 Combine like terms: -8 + -12 = -20 -20 + -3n + 4n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -20 + -3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -5 + -0.75n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.75n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.75n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.75n + n2 = 0 + 5 -0.75n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.75n + n2 = 5 The n term is -0.75n. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75n + 0.140625 + n2 = 5 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 5 + 0.140625 Combine like terms: 5 + 0.140625 = 5.140625 0.140625 + -0.75n + n2 = 5.140625 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 5.140625 Calculate the square root of the right side: 2.267294643 Break this problem into two subproblems by setting (n + -0.375) equal to 2.267294643 and -2.267294643.Subproblem 1
n + -0.375 = 2.267294643 Simplifying n + -0.375 = 2.267294643 Reorder the terms: -0.375 + n = 2.267294643 Solving -0.375 + n = 2.267294643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 2.267294643 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 2.267294643 + 0.375 n = 2.267294643 + 0.375 Combine like terms: 2.267294643 + 0.375 = 2.642294643 n = 2.642294643 Simplifying n = 2.642294643Subproblem 2
n + -0.375 = -2.267294643 Simplifying n + -0.375 = -2.267294643 Reorder the terms: -0.375 + n = -2.267294643 Solving -0.375 + n = -2.267294643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -2.267294643 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -2.267294643 + 0.375 n = -2.267294643 + 0.375 Combine like terms: -2.267294643 + 0.375 = -1.892294643 n = -1.892294643 Simplifying n = -1.892294643Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.642294643, -1.892294643}
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