square, square, square roots!

Bnow wrote:Answer is 5, but I'm not sure I totally see how. If I plug in, and square 5, I get 25, then minus 2 to get square root 23, then square that to get 529 minus 1, to get 528, and then square that...? Argh. Feel like I should get this right away

If you plug in the answer choices, you must do it in place of x. Why are you squaring 5 and subtracting 2 from it! If you plug 5 for x the expression will look like: √(1 - √(2 - √5)), which is not real as √5 > 2, and thus √(2 - √5) is an imaginary number. I believe the posted question is not the original one (otherwise 5 shouldn't be the answer and other fours will be). I've made the necessary changes in the question quoted below.

For which of the following values of x, √(1 - √(2 - √x)) is NOT defined as a real number?

• 1
  2
  3
  4
  5

Though plugging the options is a very effective way to solve this problem, I want to provide some insight.

An expression containing square roots will be a real number only if there are no negative numbers under the square root. The expression given has three square roots in it.

For √(1 - √(2 - √x)) to be real (1 - √(2 - √x)) shouldn't be negative, i.e. must be greater than or equal to zero.

i.e. (1 - √(2 - √x)) ≥ 0
=> 1 ≥ √(2 - √x)

Now for √(2 - √x) to be real (2 - √x) shouldn't be negative, i.e. must be greater than or equal to zero. Combining this new condition with the above one we get,
=> 0 ≤ (2 - √x) ≤ 1
=> 1 ≤ √x ≤ 2
=> 1 ≤ x ≤ 4

Thus for the expression to be NOT real x must be less than 1 or greater than 4. Only such option is 5.

The correct answer is E.

gmat7202011 wrote:I think this expression will be real for a value of x =0, isn't it. Am i missing something

Thanks again

Put x = 0 in the expression: √(1 - √(2 - √0)) = √(1 - √2) = √(1 - 1.414) = √(-0.414) = NOT real