$\begin{align} & \text{If the roots of the equation }{{x}^{2}}+7x+8=0\text{ are whose roots are }\alpha \text{ and }\beta ,\text{ } \\ & \text{Find the equation whose roots are }{{\alpha }^{2}}\text{ and }{{\beta }^{2}} \\\end{align}$
$\begin{align} & \text{The given equation is }{{x}^{2}}+7x+8=0,\text{ whose roots are given }\alpha \text{ and }\beta \\ & a=1,b=7,c=8 \\ & \text{Sum of roots}=(\alpha +\beta )=-\frac{b}{a}=-\frac{7}{1}=-7 \\ & \text{Product of roots}=\alpha \beta =\frac{c}{a}=\frac{8}{1}=8 \\ & \text{The new equation we want to form }{{\alpha }^{2}}\text{ and }{{\beta }^{2}} \\ & \text{Sum of new roots }{{\alpha }^{2}}+{{\beta }^{2}}={{(\alpha +\beta )}^{2}}-2\alpha \beta \\ & {{\alpha }^{2}}+{{\beta }^{2}}={{(-7)}^{2}}-2(8)=49-16=33 \\ & \text{Product of new roots}={{\alpha }^{2}}{{\beta }^{2}}={{(\alpha \beta )}^{2}}={{8}^{2}}=64 \\ & \text{The new equation form will be of the form} \\ & {{x}^{2}}-(\text{sum of new roots)}x+\text{ }(\text{product of new roots)}=0 \\ & {{x}^{2}}-33x+64=0 \\\end{align}$