PyjamaScientist wrote:
Square both sides
I got stuck on this approach. I know the correct solution now, but why did this simple operation of squaring both side give me such erroneous results?
Is it a rule that you shouldn't square a square root on GMAT or it might screw up your analysis?
The fundamental issue here is that the "√" square root
symbol does not mean exactly the same thing as the phrase "square root". The number 4, for example, has two square roots, 2 and -2, because if you square either of those numbers, you get 4. But the square root symbol "√" means
only the positive square root (or, technically, the non-negative square root if you might be taking the root of zero). Since we must get something positive (or zero) when we apply the "√" to something, we sometimes have to account for that before doing anything else. So if you have this equation:
a = √1
there's obviously no need to do anything to solve, because √1 = 1, so here a = 1 is obviously the only solution. But if instead we decided to solve by squaring both sides, we'd get:
a^2 = 1
and from this new equation, you might think a = 1 and a = -1 are both valid solutions. And they would be if the symbol "√" meant "either the positive and the negative square root", but that's not what it means. It is only equal to one of those two roots. So when we see "a = √(something)", we first need to recognize that a must be at least zero before doing anything else, and then after we square and solve, we need to discard any solutions that are negative. To give a non-trivial example, if you see, say, z^3 = √(z^2), then you might indeed want to square both sides, but before you do that, you have to recognize that z^3, and therefore z, must be 0 or greater, or else you might solve and think that z can equal -1, and it can't. It can only equal 0 or 1 in this case.
Applying that to the GMAT question in this thread, we have this equation:
3 - x = √(x - 3)^2
and since 3 - x is equal to "√(something)", we first need to recognize that 3 - x must be at least zero. So 3 - x
> 0, and x
< 3. Now you can go ahead and square both sides and solve as you did, and when you arrive at "0 = 0", that just means the equation is always true, for any valid values. Since we know in advance x
< 3 must be true, we then can correctly conclude that the equation is true for every value of x that is 3 or less.
That's not how I'd solve this question -- instead we can notice that √(x - 3)^2 = |x - 3| = |3 - x|. So the question is asking: is 3 - x = |3 - x|? If that equation is true, that means the absolute value isn't doing anything (it's not changing "3 - x" into something else; it's leaving it completely unchanged), so this equation is only going to be true if 3 - x is positive or zero, or in other words if x
< 3. From that point on, analyzing the statements is straightforward.
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