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Simplifying 4n2 + -3n + -4 = -2 Reorder the terms: -4 + -3n + 4n2 = -2 Solving -4 + -3n + 4n2 = -2 Solving for variable 'n'. Reorder the terms: -4 + 2 + -3n + 4n2 = -2 + 2 Combine like terms: -4 + 2 = -2 -2 + -3n + 4n2 = -2 + 2 Combine like terms: -2 + 2 = 0 -2 + -3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + -0.75n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.75n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.75n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.75n + n2 = 0 + 0.5 -0.75n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.75n + n2 = 0.5 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 = 0.5 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 0.5 + 0.140625 Combine like terms: 0.5 + 0.140625 = 0.640625 0.140625 + -0.75n + n2 = 0.640625 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (n + -0.375) equal to 0.80039053 and -0.80039053.Subproblem 1
n + -0.375 = 0.80039053 Simplifying n + -0.375 = 0.80039053 Reorder the terms: -0.375 + n = 0.80039053 Solving -0.375 + n = 0.80039053 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 = 0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 0.80039053 + 0.375 n = 0.80039053 + 0.375 Combine like terms: 0.80039053 + 0.375 = 1.17539053 n = 1.17539053 Simplifying n = 1.17539053Subproblem 2
n + -0.375 = -0.80039053 Simplifying n + -0.375 = -0.80039053 Reorder the terms: -0.375 + n = -0.80039053 Solving -0.375 + n = -0.80039053 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 = -0.80039053 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -0.80039053 + 0.375 n = -0.80039053 + 0.375 Combine like terms: -0.80039053 + 0.375 = -0.42539053 n = -0.42539053 Simplifying n = -0.42539053Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.17539053, -0.42539053}
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