x, y - the numbers
The numbers multiply to 100.
\(xy=100\)
The numbers add up to -29.
\(x+y=-29 \\
y=-x-29\)
Substitute -x-29 for y in the first equation:
\(x(-x-29)=100 \\
-x^2-29x=100 \\
-x^2-29x-100=0 \\
-x^2-4x-25x-100=0 \\
-x(x+4)-25(x+4)=0 \\
(-x-25)(x+4)=0 \\
-x-25=0 \ \lor \ x+4=0 \\
x=-25 \ \lor \ x=-4 \\ \\
y=-x-29 \\
y=-(-25)-29 \ \lor \ y=-(-4)-29 \\
y=25-29 \ \lor \ y=4-29 \\
y=-4 \ \lor \ y=-25 \\ \\
(x,y)=(-25,-4) \hbox{ or } (x,y)=(-4,-25)\)
The numbers are -4 and -25.
