Y=85+.5Y(-40+0.25Y)+85+60

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Solution for Y=85+.5Y(-40+0.25Y)+85+60 equation:


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 = 156.221880341

Subproblem 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 = 11.778119659

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {156.221880341, 11.778119659}

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