Question

Find two numbers whose difference is 13 and whose squares when added is 449.

Answer 1

X, y - the numbers

$x-y=13 \\ x^2+y^2=449 \\ \\ \hbox{solve the first equation for x:} \\ x-y=13 \\ x=13+y \\ \\ \hbox{substitute the result for x in the second equation:} \\ (13+y)^2+y^2=449 \\ 169+26y+y^2+y^2=449 \\ 2y^2+26y+169-449=0 \\ 2y^2+26y-280=0 \ \ \ \ \ \ \ \ \ \ \ |\div 2 \\ y^2+13y-140=0 \\ y^2+20y-7y-140=0 \\ y(y+20)-7(y+20)=0 \\ (y-7)(y+20)=0 \\ y-7=0 \ \lor \ y+20=0 \\ y=7 \ \lor \ y=-20$

$x=13+y \\ \Downarrow \\ \hbox{for } y=7: \\ x=13+7 \\ x=20 \\ \\ \hbox{for } y=-20: \\ x=13-20 \\ x=-7 \\ \\ (x,y)=(20,7) \hbox{ or } (x,y)=(-7,-20)$

The numbers are 20 and 7 or -7 and -20.