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 * 1 * (-2250) =25+9000=9025 \\ x=(-b \pm √(b^2-4ac))/(2a)=(-5 \pm √(9025))/(2 * 1)=(-5 \pm 95)/(2) \\ x=(-5 - 95)/(2) \ \lor \ x=(-5+95)/(2) \\ x=(-100)/(2) \ \lor \ x=(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.