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Simplifying 5(-8 + 5n) + -4n(n + -4) = 0 (-8 * 5 + 5n * 5) + -4n(n + -4) = 0 (-40 + 25n) + -4n(n + -4) = 0 Reorder the terms: -40 + 25n + -4n(-4 + n) = 0 -40 + 25n + (-4 * -4n + n * -4n) = 0 -40 + 25n + (16n + -4n2) = 0 Combine like terms: 25n + 16n = 41n -40 + 41n + -4n2 = 0 Solving -40 + 41n + -4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 10 + -10.25n + n2 = 0 Move the constant term to the right: Add '-10' to each side of the equation. 10 + -10.25n + -10 + n2 = 0 + -10 Reorder the terms: 10 + -10 + -10.25n + n2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + -10.25n + n2 = 0 + -10 -10.25n + n2 = 0 + -10 Combine like terms: 0 + -10 = -10 -10.25n + n2 = -10 The n term is -10.25n. Take half its coefficient (-5.125). Square it (26.265625) and add it to both sides. Add '26.265625' to each side of the equation. -10.25n + 26.265625 + n2 = -10 + 26.265625 Reorder the terms: 26.265625 + -10.25n + n2 = -10 + 26.265625 Combine like terms: -10 + 26.265625 = 16.265625 26.265625 + -10.25n + n2 = 16.265625 Factor a perfect square on the left side: (n + -5.125)(n + -5.125) = 16.265625 Calculate the square root of the right side: 4.033066451 Break this problem into two subproblems by setting (n + -5.125) equal to 4.033066451 and -4.033066451.Subproblem 1
n + -5.125 = 4.033066451 Simplifying n + -5.125 = 4.033066451 Reorder the terms: -5.125 + n = 4.033066451 Solving -5.125 + n = 4.033066451 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + n = 4.033066451 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + n = 4.033066451 + 5.125 n = 4.033066451 + 5.125 Combine like terms: 4.033066451 + 5.125 = 9.158066451 n = 9.158066451 Simplifying n = 9.158066451Subproblem 2
n + -5.125 = -4.033066451 Simplifying n + -5.125 = -4.033066451 Reorder the terms: -5.125 + n = -4.033066451 Solving -5.125 + n = -4.033066451 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + n = -4.033066451 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + n = -4.033066451 + 5.125 n = -4.033066451 + 5.125 Combine like terms: -4.033066451 + 5.125 = 1.091933549 n = 1.091933549 Simplifying n = 1.091933549Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.158066451, 1.091933549}
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