a,b,c are real.Let alpha,beta be the roots of ax^2+bx+c=0.Express the roots of a^3x^2+abcx+c^3=0 in terms of alpha and beta.a is not equal to zero

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0 \\ \mathrm{\alpha }+\mathrm{\beta }=-\frac{\mathrm{b}}{\mathrm{a}} \\ \mathrm{\alpha \beta }=\frac{\mathrm{c}}{\mathrm{a}} \\ {\mathrm{a}}^3{\mathrm{x}}^2+\mathrm{abcx}+{\mathrm{c}}^3=0 \\ \mathrm{Divide} \mathrm{by} {\mathrm{a}}^3 \\ \Rightarrow {\mathrm{x}}^2+\frac{\mathrm{abc}}{{\mathrm{a}}^3}\mathrm{x}+{\left(\frac{\mathrm{c}}{\mathrm{a}}\right)}^3=0 \\ \Rightarrow {\mathrm{x}}^2+\left(\frac{\mathrm{b} }{\mathrm{a}}\right)\left(\frac{\mathrm{c}}{\mathrm{a}}\right)\mathrm{x}+{\left(\frac{\mathrm{c}}{\mathrm{a}}\right)}^3=0 \\ \Rightarrow {\mathrm{x}}^2-\left(\mathrm{\alpha }+\mathrm{\beta }\right)\mathrm{\alpha \beta x}+{\left(\mathrm{\alpha \beta }\right)}^3=0 \\ \Rightarrow {\mathrm{x}}^2-\left({\mathrm{\alpha }}^2\mathrm{\beta }+{\mathrm{\alpha \beta }}^2\right)\mathrm{x}+\left({\mathrm{\alpha }}^2\mathrm{\beta }\right).\left({\mathrm{\alpha \beta }}^2\right)=0 \\ \mathrm{On} \mathrm{comparing} \mathrm{with} {\mathrm{x}}^2-\left(\mathrm{Sum} \mathrm{of} \mathrm{roots}\right)\mathrm{x}+\mathrm{Product} \mathrm{of} \mathrm{roots}=0, \mathrm{we} \mathrm{get} \mathrm{that} \\ \mathbf{roots}\mathbf{ }\mathbf{are}\mathbf{ }{\mathbf{\alpha }}^{\mathbf{2}}\mathbf{\beta }\mathbf{,}{\mathbf{\alpha \beta }}^{\mathbf{2}} \end{gathered}$$

Hope this information will clear your doubts about this topic.

If you have any doubts just ask here on the ask and answer forum and our experts will try to help you out as soon as possible.
Regards