x, y - the numbers
The numbers add up to 5 and multiply to -36.
[tex]x+y=5 \\
xy=-36 \\ \\
y=5-x \\
xy=-36 \\ \\
\hbox{substitute 5-x for y in the second equation:} \\
x(5-x)=-36 \\
5x-x^2=-36 \\
-x^2+5x+36=0 \\
-x^2-4x+9x+36=0 \\
-x(x+4)+9(x+4)=0 \\
(-x+9)(x+4)=0 \\
-x+9=0 \ \lor \ x+4=0 \\
x=9 \ \lor \ x=-4 \\ \\
y=5-x \\
y=5-9 \ \lor \ y=5-(-4) \\
y=-4 \ \lor \ y=9 \\ \\
(x,y)=(9,-4) \hbox{ or } (x,y)=(-4,9)[/tex]
The numbers are -4 and 9.
