uppose a machine with a floating point system $(\backslash$beta $,$ t$,$ L$,$U $)=(10,8,50,50)$ is used to cal$-$

culate the roots of the quadratic equation

ax$2+$ bx $+$ c $=0,$

where a$,$ b$,$ and c are given, real coefficients.

For each of the following, state the numerical difficulties that arise if one uses the standard

formula for computing the roots. Explain how to overcome these difficulties $($when possible$).$

$($a$)$ a $=1$ ; b $=105$ ; c $=1.$

$($b$)$ a $=61030$ ; b $=51030$ ; c $=41030.$

$($c$)$ a $=1030$ ; b $=1030$ ; c $=1030$