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Simplifying 500 = 1000 + 5n + -4n2 Solving 500 = 1000 + 5n + -4n2 Solving for variable 'n'. Combine like terms: 500 + -1000 = -500 -500 + -5n + 4n2 = 1000 + 5n + -4n2 + -1000 + -5n + 4n2 Reorder the terms: -500 + -5n + 4n2 = 1000 + -1000 + 5n + -5n + -4n2 + 4n2 Combine like terms: 1000 + -1000 = 0 -500 + -5n + 4n2 = 0 + 5n + -5n + -4n2 + 4n2 -500 + -5n + 4n2 = 5n + -5n + -4n2 + 4n2 Combine like terms: 5n + -5n = 0 -500 + -5n + 4n2 = 0 + -4n2 + 4n2 -500 + -5n + 4n2 = -4n2 + 4n2 Combine like terms: -4n2 + 4n2 = 0 -500 + -5n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -125 + -1.25n + n2 = 0 Move the constant term to the right: Add '125' to each side of the equation. -125 + -1.25n + 125 + n2 = 0 + 125 Reorder the terms: -125 + 125 + -1.25n + n2 = 0 + 125 Combine like terms: -125 + 125 = 0 0 + -1.25n + n2 = 0 + 125 -1.25n + n2 = 0 + 125 Combine like terms: 0 + 125 = 125 -1.25n + n2 = 125 The n term is -1.25n. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25n + 0.390625 + n2 = 125 + 0.390625 Reorder the terms: 0.390625 + -1.25n + n2 = 125 + 0.390625 Combine like terms: 125 + 0.390625 = 125.390625 0.390625 + -1.25n + n2 = 125.390625 Factor a perfect square on the left side: (n + -0.625)(n + -0.625) = 125.390625 Calculate the square root of the right side: 11.197795542 Break this problem into two subproblems by setting (n + -0.625) equal to 11.197795542 and -11.197795542.Subproblem 1
n + -0.625 = 11.197795542 Simplifying n + -0.625 = 11.197795542 Reorder the terms: -0.625 + n = 11.197795542 Solving -0.625 + n = 11.197795542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = 11.197795542 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = 11.197795542 + 0.625 n = 11.197795542 + 0.625 Combine like terms: 11.197795542 + 0.625 = 11.822795542 n = 11.822795542 Simplifying n = 11.822795542Subproblem 2
n + -0.625 = -11.197795542 Simplifying n + -0.625 = -11.197795542 Reorder the terms: -0.625 + n = -11.197795542 Solving -0.625 + n = -11.197795542 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + n = -11.197795542 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + n = -11.197795542 + 0.625 n = -11.197795542 + 0.625 Combine like terms: -11.197795542 + 0.625 = -10.572795542 n = -10.572795542 Simplifying n = -10.572795542Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.822795542, -10.572795542}
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