x, y - the numbers
\(xy=-16 \\
x+y=15 \\ \\
\hbox{solve the second equation for y:} \\
x+y=15 \\
y=15-x \\ \\
\hbox{substitute 15-x for y in the first equation:} \\
x(15-x)=-16 \\
15x-x^2=-16 \\
-x^2+15x+16=0 \\
-x^2-x+16x+16=0 \\
-x(x+1)+16(x+1)=0 \\
(-x+16)(x+1)=0 \\
-x+16=0 \ \lor \ x+1=0 \\
x=16 \ \lor \ x=-1 \\ \\
y=15-x \\
y=15-16 \ \lor \ y=15-(-1) \\
y=-1 \ \lor \ y=16 \\ \\
(x,y)=(16,-1) \hbox{ or } (x,y)=(-1,16)\)
The numbers are -1 and 16.
