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Simplifying 5n2 + -2n + -5 = 0 Reorder the terms: -5 + -2n + 5n2 = 0 Solving -5 + -2n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + -0.4n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.4n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -0.4n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.4n + n2 = 0 + 1 -0.4n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.4n + n2 = 1 The n term is -0.4n. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4n + 0.04 + n2 = 1 + 0.04 Reorder the terms: 0.04 + -0.4n + n2 = 1 + 0.04 Combine like terms: 1 + 0.04 = 1.04 0.04 + -0.4n + n2 = 1.04 Factor a perfect square on the left side: (n + -0.2)(n + -0.2) = 1.04 Calculate the square root of the right side: 1.019803903 Break this problem into two subproblems by setting (n + -0.2) equal to 1.019803903 and -1.019803903.Subproblem 1
n + -0.2 = 1.019803903 Simplifying n + -0.2 = 1.019803903 Reorder the terms: -0.2 + n = 1.019803903 Solving -0.2 + n = 1.019803903 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = 1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = 1.019803903 + 0.2 n = 1.019803903 + 0.2 Combine like terms: 1.019803903 + 0.2 = 1.219803903 n = 1.219803903 Simplifying n = 1.219803903Subproblem 2
n + -0.2 = -1.019803903 Simplifying n + -0.2 = -1.019803903 Reorder the terms: -0.2 + n = -1.019803903 Solving -0.2 + n = -1.019803903 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = -1.019803903 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = -1.019803903 + 0.2 n = -1.019803903 + 0.2 Combine like terms: -1.019803903 + 0.2 = -0.819803903 n = -0.819803903 Simplifying n = -0.819803903Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.219803903, -0.819803903}
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