The sum of two numbers is 1000 and the difference between their square is 256000. Find the numbers.

Given:

The sum of two numbers is 1000 and the difference between their square is 256000. 

 To do:

We have to find the numbers.

Solution:

Let one of the numbers be $x$.

This implies,

The other number $=1000-x$.

According to the question,

$x^2-(1000-x)^2=256000$ or $(1000-x)^2-x^2=256000$

$x^2-(1000000+x^2-2\times1000x)=256000$ or $1000000+x^2-2\times1000x-x^2=256000$

$x^2-x^2-1000000+2000x=256000$ or $x^2-x^2+1000000-2000x=256000$

$2000x=256000+1000000$ or $2000x=1000000-256000$

$2000x=1256000$ or $2000x=744000$

$x=\frac{1256000}{2000}$ or $x=\frac{744000}{2000}$

$x=628$ or $x=372$

This implies,

$1000-x=1000-628=372$ or $1000-372=628$

Therefore, the two numbers are $372$ and $628$.