Question

If $a,b,c$ are the roots of $x^3+qx+r=0$, form the equation whose roots are
$b^2{c}^2,c^2{a}^2,a^2{b}^2$.

Answer 1

Given equation, $x^3+qx+r=0\dots \dots \dots \dots \dots \dots \left(1\right)$

$b^2{c}^2=\frac{a^2{b}^2{c}^2}{a^2}=\frac{r^2}{a^2}$

$⟹y=\frac{r^2}{x^2}⟹x=\frac{r}{\sqrt{y}}\dots \dots \dots \dots .\left(2\right)$

Substituting value of $x$ from (2) in (1), we get

${\left(\frac{r}{\sqrt{y}}\right)}^3+q\left(\frac{r}{\sqrt{y}}\right)+r=0$

$⟹r\sqrt{y^3}+qry+r^3=0⟹\sqrt{y^3}=-(qy+r^2)=0$

$y^3=q^2{y}^2+2q{r}^{2}y+r^4\left[\text{Taking squares on both sides}\right]$

$⟹y^3-q^2{y}^2-2q{r}^{2}y-r^4=0$