Question:
The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.
Solution:
Let the larger numbers be $x$
Then according to question,
Square of the smaller number be $=8 x$ then
$x^{2}-8 x=180$
$x^{2}-8 x-180=0$
$x^{2}-18 x+10 x-180=0$
$x(x-18)+10(x-18)=0$
$(x-18)(x+10)=0$
$(x-18)=0$
$x=18$
Or
$(x+10)=0$
$x=-10$
Since, x being a positive integer so, x cannot be negative,
Therefore,
When $x=18$ then smaller number be
$\sqrt{8 x}=\sqrt{8 \times 18}$
$=\sqrt{144}$
$=\pm 12$
Thus, two consecutive number be 18,12 or $18,-12$
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