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Simplifying Y = 85 + 0.5Y(-40 + 0.25Y) + 85 + 60 Y = 85 + (-40 * 0.5Y + 0.25Y * 0.5Y) + 85 + 60 Y = 85 + (-20Y + 0.125Y2) + 85 + 60 Reorder the terms: Y = 85 + 85 + 60 + -20Y + 0.125Y2 Combine like terms: 85 + 85 = 170 Y = 170 + 60 + -20Y + 0.125Y2 Combine like terms: 170 + 60 = 230 Y = 230 + -20Y + 0.125Y2 Solving Y = 230 + -20Y + 0.125Y2 Solving for variable 'Y'. Reorder the terms: -230 + Y + 20Y + -0.125Y2 = 230 + -20Y + 0.125Y2 + -230 + 20Y + -0.125Y2 Combine like terms: Y + 20Y = 21Y -230 + 21Y + -0.125Y2 = 230 + -20Y + 0.125Y2 + -230 + 20Y + -0.125Y2 Reorder the terms: -230 + 21Y + -0.125Y2 = 230 + -230 + -20Y + 20Y + 0.125Y2 + -0.125Y2 Combine like terms: 230 + -230 = 0 -230 + 21Y + -0.125Y2 = 0 + -20Y + 20Y + 0.125Y2 + -0.125Y2 -230 + 21Y + -0.125Y2 = -20Y + 20Y + 0.125Y2 + -0.125Y2 Combine like terms: -20Y + 20Y = 0 -230 + 21Y + -0.125Y2 = 0 + 0.125Y2 + -0.125Y2 -230 + 21Y + -0.125Y2 = 0.125Y2 + -0.125Y2 Combine like terms: 0.125Y2 + -0.125Y2 = 0.000 -230 + 21Y + -0.125Y2 = 0.000 Begin completing the square. Divide all terms by -0.125 the coefficient of the squared term: Divide each side by '-0.125'. 1840 + -168Y + Y2 = 0 Move the constant term to the right: Add '-1840' to each side of the equation. 1840 + -168Y + -1840 + Y2 = 0 + -1840 Reorder the terms: 1840 + -1840 + -168Y + Y2 = 0 + -1840 Combine like terms: 1840 + -1840 = 0 0 + -168Y + Y2 = 0 + -1840 -168Y + Y2 = 0 + -1840 Combine like terms: 0 + -1840 = -1840 -168Y + Y2 = -1840 The Y term is -168Y. Take half its coefficient (-84). Square it (7056) and add it to both sides. Add '7056' to each side of the equation. -168Y + 7056 + Y2 = -1840 + 7056 Reorder the terms: 7056 + -168Y + Y2 = -1840 + 7056 Combine like terms: -1840 + 7056 = 5216 7056 + -168Y + Y2 = 5216 Factor a perfect square on the left side: (Y + -84)(Y + -84) = 5216 Calculate the square root of the right side: 72.221880341 Break this problem into two subproblems by setting (Y + -84) equal to 72.221880341 and -72.221880341.Subproblem 1
Y + -84 = 72.221880341 Simplifying Y + -84 = 72.221880341 Reorder the terms: -84 + Y = 72.221880341 Solving -84 + Y = 72.221880341 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '84' to each side of the equation. -84 + 84 + Y = 72.221880341 + 84 Combine like terms: -84 + 84 = 0 0 + Y = 72.221880341 + 84 Y = 72.221880341 + 84 Combine like terms: 72.221880341 + 84 = 156.221880341 Y = 156.221880341 Simplifying Y = 156.221880341Subproblem 2
Y + -84 = -72.221880341 Simplifying Y + -84 = -72.221880341 Reorder the terms: -84 + Y = -72.221880341 Solving -84 + Y = -72.221880341 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '84' to each side of the equation. -84 + 84 + Y = -72.221880341 + 84 Combine like terms: -84 + 84 = 0 0 + Y = -72.221880341 + 84 Y = -72.221880341 + 84 Combine like terms: -72.221880341 + 84 = 11.778119659 Y = 11.778119659 Simplifying Y = 11.778119659Solution
The solution to the problem is based on the solutions from the subproblems. Y = {156.221880341, 11.778119659}
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