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Simplifying 7n + 4n(n + -2) = 124 Reorder the terms: 7n + 4n(-2 + n) = 124 7n + (-2 * 4n + n * 4n) = 124 7n + (-8n + 4n2) = 124 Combine like terms: 7n + -8n = -1n -1n + 4n2 = 124 Solving -1n + 4n2 = 124 Solving for variable 'n'. Reorder the terms: -124 + -1n + 4n2 = 124 + -124 Combine like terms: 124 + -124 = 0 -124 + -1n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -31 + -0.25n + n2 = 0 Move the constant term to the right: Add '31' to each side of the equation. -31 + -0.25n + 31 + n2 = 0 + 31 Reorder the terms: -31 + 31 + -0.25n + n2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + -0.25n + n2 = 0 + 31 -0.25n + n2 = 0 + 31 Combine like terms: 0 + 31 = 31 -0.25n + n2 = 31 The n term is -0.25n. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25n + 0.015625 + n2 = 31 + 0.015625 Reorder the terms: 0.015625 + -0.25n + n2 = 31 + 0.015625 Combine like terms: 31 + 0.015625 = 31.015625 0.015625 + -0.25n + n2 = 31.015625 Factor a perfect square on the left side: (n + -0.125)(n + -0.125) = 31.015625 Calculate the square root of the right side: 5.569167352 Break this problem into two subproblems by setting (n + -0.125) equal to 5.569167352 and -5.569167352.Subproblem 1
n + -0.125 = 5.569167352 Simplifying n + -0.125 = 5.569167352 Reorder the terms: -0.125 + n = 5.569167352 Solving -0.125 + n = 5.569167352 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 5.569167352 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 5.569167352 + 0.125 n = 5.569167352 + 0.125 Combine like terms: 5.569167352 + 0.125 = 5.694167352 n = 5.694167352 Simplifying n = 5.694167352Subproblem 2
n + -0.125 = -5.569167352 Simplifying n + -0.125 = -5.569167352 Reorder the terms: -0.125 + n = -5.569167352 Solving -0.125 + n = -5.569167352 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -5.569167352 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -5.569167352 + 0.125 n = -5.569167352 + 0.125 Combine like terms: -5.569167352 + 0.125 = -5.444167352 n = -5.444167352 Simplifying n = -5.444167352Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.694167352, -5.444167352}
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