Find the roots of the equation (x-a)(x-b)=(x-c)(x-d), if you know that a+d=b+c=2020 and a≠c.
Dividing by (x-b) and (x-d) gives:
(x-a)/(x-d) = (x-c)/(x-b)
By inspection, this has no solution if a+d=b+c=2020 and a≠c.
So, the only solution is if (x-d) = 0 or (x-b) = 0.
If x-b = 0, then either x-d or x-c =0. But x-d cannot also = 0, because then a = c
So, b = c = 1010 = x
This gives rise to solutions (a,b,c,d,x) = (a, 1010, 1010, 2020-a, 1010).
Similarly, if x-d = 0, solutions are (1010, b, 2020-b, 1010,1010).
In either case, the only possible x value = 1010
Solution
