9.2. Petya says to Kolya: «If you think of a quadratic trinomial that has roots and tell me only the leading coefficient and the distance between the roots, then I will guess the y-coordinate of the vertex on its graph». Kolya believes that Petya is wrong: after all, to define a quadratic trinomial, you need to know three numbers. Who is right?
Answer: Pete. Solution. Let $a$ be the leading coefficient, $d=\left|x_{2}-x_{1}\right|$ be the distance between the roots. The result follows from the relations $a x^{2}+b x+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)=$ $a\left(\left(x-\frac{x_{1}+x_{2}}{2}\right)^{2}+x_{1} x_{2}-\left(\frac{x_{1}+x_{2}}{2}\right)^{2}\right)=a\left(\left(x-\frac{x_{1}+x_{2}}{2}\right)^{2}-\left(\frac{d}{2}\right)^{2}\right)$. Thus, the ordinate of the vertex is $-a\left(\frac{d}{2}\right)^{2}$ and depends only on $a$ and $d$.