about roots of an equation

In the quadratic equation $ax^2+bx+c=0$ the sum of the roots is $\dfrac{-b}{a}$ and the product of the roots is $\dfrac{c}{a}$ .

$\alpha + \beta = \dfrac{-(-p)}{1}=p$

$\alpha \times \beta =\dfrac{q}{1}=q$

Then,

$(\alpha+\beta)^2=1^2$

$\alpha^2+2\alpha \beta +\beta^2=p^2$

$\alpha^2 + \beta^2=p^2-2\alpha \beta$

however, $\alpha \beta = q$

$\alpha^2 + \beta^2 = \boxed{p^2-2q}$