So the questions says -
Let f(x),g(x) and h(x) be quadratic polynomials having positive leading coefficients and real and distinct roots. If each pair of them has a common root, then find roots of f(x)+g(x)+h(x)=0.
What I did -
Let,
f(x)=a1(x−α)(x−β),g(x)=a2(x−β)(x−γ),h(x)=a3(x−γ)(x−α),F(x):=f(x)+g(x)+h(x)
Now,
F(α)=a2(α−β)(α−γ)F(β)=a3(β−γ)(β−α)F(γ)=a1(γ−α)(γ−β)
I don't know how to proceed further. I referred to the solution, it just multiplies F(α),F(β), and F(γ) and it comes out to be negative. And hence it concludes that roots of F(x)=0 are real and distinct. Can anyone explain why?
Thanks.
Answer
Suppose, without loss of generality, that α<β<γ: it is easy to check that F(α)>0, F(β)<0 and F(γ)>0. But F(x) is a quadratic polynomial, hence a continuous function: it follows that F(x)=0 for some x between α and β, and also for some x between β and γ.
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