x, y - the numbers
The product of the numbers is 2250. Their difference is 5.
[tex]xy=2250 \\
y-x=5 \\ \\
xy=2250 \\
y=5+x \\ \\
\hbox{substitute 5+x for y in the first equation:} \\
x(5+x)=2250 \\
5x+x^2=2250 \\
x^2+5x-2250=0 \\ \\
a=1 \\ b=5 \\ c=-2250 \\
b^2-4ac=5^2-4 \times 1 \times (-2250) =25+9000=9025 \\
x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}=\frac{-5 \pm \sqrt{9025}}{2 \times 1}=\frac{-5 \pm 95}{2} \\
x=\frac{-5 - 95}{2} \ \lor \ x=\frac{-5+95}{2} \\
x=\frac{-100}{2} \ \lor \ x=\frac{90}{2} \\
x=-50 \ \lor \ x=45 \\ \\
y=5+x \\
y=5-50 \ \lor \ y=5+45 \\
y=-45 \ \lor \ y=50[/tex]
[tex](x,y)=(-50,-45) \hbox{ or } (x,y)=(45,50)[/tex]
The numbers are -50 and -45 or 45 and 50.
