Mathematics - Quadratic Equation Question with Solution | TestHub

For the equation $x^2+px+q=0$, we have $\alpha +\beta =-p$ and $\alpha \beta =q$. Let the new roots be $y_1=\frac{\alpha }{\beta }$ and $y_2=\frac{\beta }{\alpha }$. The sum of the new roots is $y_1+y_2=\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }=\frac{(-p{)}^2-2q}{q}=\frac{p^2-2q}{q}$. The product of the new roots is $y_1{y}_2=\frac{\alpha }{\beta }\cdot \frac{\beta }{\alpha }=1$. The required quadratic equation is $x^2-(y_1+y_2)x+y_1{y}_2=0$, which is $x^2-\left(\frac{p^2-2q}{q}\right)x+1=0$. Multiplying by $q$, we get $q{x}^2-(p^2-2q)x+q=0$.