It is a question from algebraQuadratic equationsOne or more than one answerMultiple answer

let f(x)= (x-a)(x-c)+lambda* (x-b)(x-d)

now, f(a)>0

f(b)

so, at least one root lies between a and b.

f(d)>0

so, at least one root lies between b and d.

but f(x) is a quadratic polynomial and cannot have more than 2 roots.

therefore both roots are real and distinct

Regards

Arun