Mathematics - Quadratic Equation Question with Solution | TestHub

For the equation $x^2-px+q=0$, we have $\alpha +\beta =p$ and $\alpha \beta =q$. For the new equation with roots ${\alpha }^2$ and ${\beta }^2$, the sum of roots is ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta =p^2-2q$. The product of roots is ${\alpha }^2{\beta }^2=(\alpha \beta {)}^2=q^2$. The new equation is $x^2-(\text{sum of roots})x+(\text{product of roots})=0$, which is $x^2-(p^2-2q)x+q^2=0$.