If we have roots of the function $y=a{x}^{2}+bx+c$ we can calculate $S=\frac{-b}{a}$ and also $P=\frac{c}{a}$. Then we know that we can form the function this way:

$${x}^{2}-Sx+P$$

So on the other side we know that we have the function $f(x)=y$ in different ways:

$$y=a{x}^{2}+bx+c$$

($\alpha $ and $\beta $ are roots of the quadratic function)

$y=a(x-\alpha )(x-\beta )$

And my question is here:

$$y=a({x}^{2}-Sx+P)$$

Actually know that how we can form the qudratic equation using ${x}^{2}-Sx+P$, but the function must be like $y=a({x}^{2}-Sx+P)$. Actually I don't know that why we add a. I know it will be removed when $(a)(\frac{-b}{a})$. But I don't know that what is $y=a({x}^{2}-Sx+P)$ different whitout a!