Mathematics - Quadratic Equation Question with Solution | TestHub

Let the roots of $x^2-px+q=0$ be $\alpha ,\beta$. Then $\alpha +\beta =p$ and $\alpha \beta =q$. The new roots are $A={\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta =p^2-2q$ and $B=\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{{\alpha }^2+{\beta }^2}{(\alpha \beta {)}^2}=\frac{p^2-2q}{q^2}$. The sum of new roots is $S=A+B=(p^2-2q)(1+\frac{1}{q^2})=(p^2-2q)\frac{q^2+1}{q^2}$. The product of new roots is $P=AB=\frac{(p^2-2q{)}^2}{q^2}$. The required equation is $x^2-Sx+P=0$, which is $q^2{x}^2-(p^2-2q)(q^2+1)x+(p^2-2q{)}^2=0$.