Without solving for x and y, determine the possible values of x/y, whenever:
x+y=29, and:
x*y=198
The system is symmetric in x and y, so if z1=x/y is a possible answer then z2=y/x also is a possible answer. z1 and z2 are the roots of (z-x/y)*(z-y/x)=0. Multiplied out we have z^2 - ((x^2+y^2)/(xy))*z + 1 = 0.
x^2+y^2 = (x+y)^2-2*(xy). Substitute this identity into our quadratic and simplify a bit: z^2 - ((x+y)^2/(xy)-2)*z + 1 = 0. Now the coefficients are in terms of x+y and xy.
Then just substitute x+y=29 and xy=198: z^2 - (29^2/(198)-2)*z + 1 = 0. Clearing fractions reduces the quadratic into 198z^2 - 445z + 198 = 0 Now just plug this in to the quadratic formula to get z = 11/18 or 18/11 as the possible values of x/y.
Solution
