Question

The sum of the squares of two positive integers is $208$. If the square of the larger number is $18$ times the smaller number, find the numbers.

Answer 1

Let the smaller number be $x$. Then,

Square of larger number $=18x$

Also, Square of the smaller number $=x^2$

It is given that the sum of the square of the integers is $208$.

$\therefore x^2+18x=208$

$\Rightarrow x^2+18x-208=0$

$\Rightarrow x^2+26x-8x-208=0$

$\Rightarrow (x+26)(x-8)=0$

$\Rightarrow x=8,x=-26$

But, the number are positive. Therefore, $x=8$

$\therefore$ Square of the larger number $=18x=18\times 8=144$

$\Rightarrow$ Larger number $=\sqrt{144}=12$

Hence, the numbers are $8$ and $12$.