The difference of the squares of two positive integers is 180. The square of the smaller number is 8 times the larger, find the numbers.
Let the larger number be x.
Then according to the question,
Square of the smaller number = 8x, then
$x^{2}-8 x=180$
$\Rightarrow x^{2}-8 x-180=0$
$\Rightarrow x^{2}-18 x+10 x-180=0$
$\Rightarrow x(x-18)+10(x-18)=0$
$\Rightarrow(x+10)(x-18)=0$
$\Rightarrow x+10=0$ or $x-18=0$
$\Rightarrow x=-10$ or $x=18$
Since, x being a positive integer so, x cannot be negative,
Therefore, larger number = 18.
then the smaller number $=\sqrt{8 \times 18}=12$
Thus, the two positive numbers are 12 and 18 .
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