Interesting question you pose..

But note that only numbers of the form XXX.5 when multiplied with 2 will give you an integer (i.e have '0.5' after the decimal point multiplied by 2 would end in .0)

And in such case, the number (XXX.5) when squared will always result in a number of the type (XXXX.25)

If you take 1.5 - 1.5^2 = 2.25.

But we are given X is an integer in the question stem. SO any number of the type XXXX.25 is not an integer. Hence if square root of (4x) is integer, we can be certain to deduce 'x' will be a perfect square and hence Sqrt(x) will be an integer.

The problem with your reasoning about the square root of x able to be 1.5 is that the stem defines x as an integer 1.5(1.5) = 2.25 (not an integer)- which integer would yield a square root that ends in a .5? The decimal has to be .5 for 2(sqrtx) to be an integer. When you square any number that ends in a 0.5 you will always get a number with 0.25 as an end, which is not an integer.

@Redhorse

https://en.wikipedia.org/wiki/Irrational ... uare_roots

Open the above link and read the first couple of lines. The square root of a non-perfect square is ALWAYS irrational. So having a sq root of 1.5 is impossible.