Suppose a machine with a floating-point system (, p, L, U) = (10, 6, 30, 20) is used to calculate 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 cases, state the numerical difficulties that arise if one uses the standard formula for computing the roots. Explain how one can overcome these difficulties (when possible):

a = 10^5 , b = 10^12, c = 10^6

a = 10, b = 10^4 , c = 1

a = 2, b = 8, c = 8.00001