the roots of 3x^2 -2x-5=0 are alpha and beta find the equation whose roots are alpha^2 and beta^2 (Posted for sacha)

$${\mathtt{3}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{5}} = {\mathtt{0}} = \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{{\mathtt{5}}}{{\mathtt{3}}}}\\
{\mathtt{x}} = -{\mathtt{1}}\\
\end{array} \right\}$$

So alpha = 5/3 and beta = -1

To get the (quadratic) equation whose roots are alpha² and beta² we construct:

(x - alpha²)*(x-beta²) = 0   or:

 x² -(alpha²+beta²)*x + alpha²*beta² = 0

 

I'll leave you to put the numbers in.