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Simplifying 4n2 + 3n + -4 = 5 Reorder the terms: -4 + 3n + 4n2 = 5 Solving -4 + 3n + 4n2 = 5 Solving for variable 'n'. Reorder the terms: -4 + -5 + 3n + 4n2 = 5 + -5 Combine like terms: -4 + -5 = -9 -9 + 3n + 4n2 = 5 + -5 Combine like terms: 5 + -5 = 0 -9 + 3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + 0.75n + n2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 0.75n + 2.25 + n2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 0.75n + n2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 0.75n + n2 = 0 + 2.25 0.75n + n2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 0.75n + n2 = 2.25 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 = 2.25 + 0.140625 Reorder the terms: 0.140625 + 0.75n + n2 = 2.25 + 0.140625 Combine like terms: 2.25 + 0.140625 = 2.390625 0.140625 + 0.75n + n2 = 2.390625 Factor a perfect square on the left side: (n + 0.375)(n + 0.375) = 2.390625 Calculate the square root of the right side: 1.54616461 Break this problem into two subproblems by setting (n + 0.375) equal to 1.54616461 and -1.54616461.Subproblem 1
n + 0.375 = 1.54616461 Simplifying n + 0.375 = 1.54616461 Reorder the terms: 0.375 + n = 1.54616461 Solving 0.375 + n = 1.54616461 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 = 1.54616461 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = 1.54616461 + -0.375 n = 1.54616461 + -0.375 Combine like terms: 1.54616461 + -0.375 = 1.17116461 n = 1.17116461 Simplifying n = 1.17116461Subproblem 2
n + 0.375 = -1.54616461 Simplifying n + 0.375 = -1.54616461 Reorder the terms: 0.375 + n = -1.54616461 Solving 0.375 + n = -1.54616461 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 = -1.54616461 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = -1.54616461 + -0.375 n = -1.54616461 + -0.375 Combine like terms: -1.54616461 + -0.375 = -1.92116461 n = -1.92116461 Simplifying n = -1.92116461Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.17116461, -1.92116461}
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