Solve 3x³ - 22x² + 48x - 72 = 0 when the roots are in H.P.

Replace x by $\frac{1}{x}$ we get $32x^3-48x^2+22x-3=0$ Let its roots are $\alpha -\beta ,\alpha ,\alpha +\beta$

Sum of the roots $=\frac{48}{32}=\frac{3}{2}$

$\begin{array}{r}3\alpha =\frac{3}{2}\\ \alpha =\frac{1}{2}\end{array}$

Also, $\alpha (\alpha +\beta )(\alpha -\beta )=\frac{3}{32}$

$\begin{array}{l}\alpha \left({\alpha }^2-{\beta }^2\right)=\frac{3}{32}\\ \frac{1}{2}\left(\frac{1}{4}-{\beta }^2\right)=\frac{3}{32}\\ \beta =\frac{1}{4}\end{array}$

Hence, the roots are

$\begin{array}{l}\frac{1}{2}-\frac{1}{4},\frac{1}{2},\frac{1}{2}+\frac{1}{4}\text{ or }\frac{1}{2}+\frac{1}{4},\frac{1}{2},\frac{1}{2}-\frac{1}{4}\\ \frac{1}{4},\frac{1}{2},\frac{3}{4}\text{ or }\frac{3}{4},\frac{1}{2},\frac{1}{4}\end{array}$

Therefore, the roots of the main equation are

$4,2,\frac{4}{3}$ or $\frac{{\displaystyle 4}}{{\displaystyle 3}},2,4$