If `a lt b lt c lt d` then show that `(x-a)(x-c)+3(x-b)(x-d)=0` has real and distinct roots.

Explanation

Let `f(x)=(x-a)(x-c)+3(x-b)(x-d)`
Then `f(a)=0+3(a-b)(a-d)gt0 [ :' a-blt0,a-dlt0]`
and `f(b)=(b-a)(b-c)+0lt0 [ :' b-agt0,b-clt0]`
Thus, one root will lie between a and b
and `f(c)=0+3(c-b)(c-d)lt0 [ :' c-bgt0,c-dlt0]`
and `f(x)=(d-a)(d-c)+0gt0 [ :' d-agt0,d-cgt0]`
Thus, one root will be between c and d. Hence roots of equation are real and distinct.