If the two numbers are x and y, xy=-1 and x+y=-17.
Since y=-1/x, x+y=x-1/x=-17,
x2-1=-17x,
x2+17x-1=0,
x2+17x=1,
x2+17x+(17/2)2=1+(17/2)2=1+289/4=293/4,
(x+(17/2))2=293/4,
x+(17/2)=±√293/2,
x=-(17/2)±√293/2, which produces two possible solutions for x"
x=-8.5±8.5586 (approx), x=0.0586 or x=-17.0586.