Difference of the squares of two numbers

let $x$ and $y$ be the numbers

$x + y = 30$ (equation 1)

$x^2 + y^2 = 458$ (equation 2)

From equation 1, $x = 30-y$ , substitute this to equation 2 and simplify.

$x^2 + y^2 = 458$

$(30-y)^2 + y^2 = 458$

$y^2 - 30y + 221 = 0$

Using the quadratic formula to solve the above equation, we have

$y = 17$

$y = 13$

From the values of $y$ above, $x$ can be $17$ or $13$ . In conclusion, the two numbers are $17$ and $13$ .

The difference of their squares is $17^2 - 13^2 = 120$ .