Sender: David Xiao Hi I have a problem here which I do not know how to solve it . Please help me thanks . Find the value of a such that 4x^{2} + 4(a2)x  8a^{2} + 14a + 31 = 0 has real roots whose sum of squares is minimum. Thank you Hi David,Consider the general quadratic Ax^{2} + Bx + C. If the roots are real then the discriminant, B^{2}  4AC must be nonnegative. In your example A = 4, B = 4(a  2) and C =  8a^{2} + 14a + 31 and the discriminant is 144(a  3)(a + 1). Thus the discriminant is nonnegative if and only if a >= 3 or a <= 1. Suppose that the roots are p and q, then p + q = B/A and pq = C/A. Thus = (p + q)^{2}  2pq = (B/A)^{2}  2(C/A) = 8(10a^{2}  22a  23) 10a^{2}  22a  23 is a quadratic with minimum at 22/(20) = 1.1. Thus this quadratic is decreasing for a less than 1.1 and increasing for a greater than 1.1. Hence the minimum is at either a = 1 or a = 3. Cheers,Harley
