Question

The prouduct of two numbers is 2250 their difference is 5 what are the two numbers

Answer 1

X, y - the numbers

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

$(x,y)=(-50,-45) \hbox{ or } (x,y)=(45,50)$

The numbers are -50 and -45 or 45 and 50.