Question

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$. Find the equation whose roots are as given below.
${\alpha }^2+{\beta }^2,\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

Answer 1

$S={\alpha }^2+{\beta }^2+\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=({\alpha }^2+{\beta }^2)(1+\frac{1}{{\alpha }^2{\beta }^2})$

$=\frac{b^2-2ac}{a^2}.\frac{a^2+c^2}{c^2}$

$P=({\alpha }^2+{\beta }^2).\frac{({\alpha }^2+{\beta }^2)}{{\alpha }^2{\beta }^2}=\frac{(b^2+2ac{)}^2}{a^2{c}^2}$

$\therefore$ Equation is :

$a^2{c}^2{x}^2-(b^2-2ac)(a^2+c^2)x+(b^2-2ac{)}^2=0$