If x^2+2αβχ -f(α)(f(α)+f(β))=0 has no real roots,find that f has got a root at (α,β)

I think that you need to clarify the question.

The phrase 'that f has got a root at (\(\alpha,\beta\))' implies that f is a function of two variables, while the equation suggests that it is a function of a single variable.

That's the question exactly as it is written.The only thing i forgot to mention is that the thing to prove is slightly different.Prove that f (which is constant) has a root at (α,β).And the equation I have posted has no real roots.If it's still unclear,just tell me whichever you think is the right solution.I really need this answer.Thanks.