Let p, q, and r be constants. One solution to the equation (x-p)(x-q) = (r-p)(r-q) is x=r. Find the other solution in terms of p, q, and r.

Expand and we have that

x^2 - (p + q)x + pq  = r^2 - (p+ q)r  + pq

x^2  - (p + q)k - r^2 + (p + q)r   = 0

The sum of the roots  =  (p + q) / 1   = p + q

The product of the roots  is [ -r^2  + (p + q)r] / 1  =  -r^2  + (p + q)r

Since  r  is one solution

Let  s   be the other   and we have

So

r + s  = p + q

s  = p + q  - r

Verify  that

r * s  =    r ( p + q - r)   =  -r^2  + (p + q)r