AIME #5 1995

[stupidityismygam]

For certain real values of [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] the equation [unparseable or potentially dangerous latex formula] has four non-real roots. The product of two of these roots is [unparseable or potentially dangerous latex formula] and the sum of the other two roots is [unparseable or potentially dangerous latex formula] where [unparseable or potentially dangerous latex formula] Find [unparseable or potentially dangerous latex formula]

[mathslug]

Interesting problem. I think I got the right answer, but I REALLY need someone to check it.

Let q, r, s, and t be the roots to x^4 + ax^3 + bx^2 + cx + d = 0.

qr = (13+i)
s+t = (3+4i)

The sum of the roots taken 2 at a time = b.

b = qr + qs + qt + rs + rt + st --> b = qr + (q+r)(s+t) + st

a,b,c, and d must be real numbers. Remember that d = (qr)(st), and since it must be real, st is equal to (13-i). B must also be real, and since qr + st produces a real number the product (q+r)(s+t) must be real as well. Therefore (q+r) = (3-4i). So...

b = (13+i) + (3-4i)(3+4i) + (13-i) = 13 + 25 + 13 = 51

b = 51

[stupidityismygam]

yep