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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
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