$${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{5}} = {\mathtt{0}} = \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{{\mathtt{5}}}{{\mathtt{3}}}}\\

{\mathtt{x}} = -{\mathtt{1}}\\

\end{array} \right\}$$

So alpha = 5/3 and beta = -1

To get the (quadratic) equation whose roots are alpha^{2} and beta^{2} we construct:

(x - alpha^{2})*(x-beta^{2}) = 0 or:

x^{2} -(alpha^{2}+beta^{2})*x + alpha^{2}*beta^{2} = 0

I'll leave you to put the numbers in.

Alan Apr 22, 2014