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

Step:Find the numbers

Let’s assume the two numbers be $x$ and $y$.

And also assume that $x$ is greater than or equal to $y$.

According to the question,

we can write the sum of the two numbers as

$x+y=1000$ ……….. (i)

Again it’s given that,

the difference between the squares of the two numbers is

$x^2–y^2=256000$

$\Rightarrow$ $(x+y)(x–y)=256000$

$\Rightarrow$ $1000(x-y)=256000$

$\Rightarrow$ $x–y=\frac{256000}{1000}$

$\Rightarrow$ $x–y=256$ ………….. (ii)

By solving (i) and (ii), we can find the two numbers

On adding the equations (i) and (ii), we get;

$(x+y)+(x-y)=1000+256$

$\Rightarrow$ $x+y+x–y=1256$

$\Rightarrow$ $2x=1256$

$\Rightarrow$ $x=\frac{1256}{2}$

$\Rightarrow$ $x=628$

Now, putting the value of $x$ in equation (i), we get

$628+y=1000$

$\Rightarrow$ $y=1000–628$

$\Rightarrow$ $y=372$

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