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# If x+sqrt(x^2-4x+4)=2, then:

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General Discussion
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Q51  V47
3
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Doing this in the mathematical way (the 3-second answer just requires you to notice that the interval answer A contains the interval in every other answer choice, so if one of the other answers were correct, A would naturally also need to be correct, and a GMAT problem cannot have two different correct answers) --

x^2 - 4x + 4 = (x-2)^2, so √(x^2 - 4x + 4) = √(x-2)^2

Many test takers, seeing something like √(a^2), will think that is equal to a. That's one of the more common errors test takers make on higher level GMAT algebra questions. It is true that √(a^2) = a when a is positive or zero. But it's not true if a is negative -- if you plug a = -3 into √(a^2), you'll see that it is equal to 3, so the sign has flipped from negative to positive. In general, all we can say is that √(a^2) = |a| is always true.

So √(x-2)^2 = |x-2|, and our equation here becomes:

x + |x - 2| = 2

We can solve this using cases:

- if the thing inside the absolute value, x-2, is positive or zero, the absolute value will do nothing, so we can erase it and solve. Then we get

x + x - 2 = 2
2x = 4
x = 2

So when x - 2 > 0, there is only one solution, x = 2.

- when x-2 is negative, though, the absolute value will flip it's sign, so it will become 2-x. So when x < 2, our equation becomes

x + 2 - x = 2
2 = 2

and the equation is always true for every value of x < 2.

Combining our solutions from the two cases, the equation will be true whenever x < 2. If that were an answer choice, it would be the right answer, but we don't find that among the choices. But if we know for certain that x < 2, then it's clearly true that x < 3, so that's the right answer here. We can't be certain any of the other answer choices are correct, because some potential values of x lie outside the intervals in every other choice.

Nice question!
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IanStewart wrote:
Doing this in the mathematical way (the 3-second answer just requires you to notice that the interval answer A contains the interval in every other answer choice, so if one of the other answers were correct, A would naturally also need to be correct, and a GMAT problem cannot have two different correct answers) --

x^2 - 4x + 4 = (x-2)^2, so √(x^2 - 4x + 4) = √(x-2)^2

Many test takers, seeing something like √(a^2), will think that is equal to a. That's one of the more common errors test takers make on higher level GMAT algebra questions. It is true that √(a^2) = a when a is positive or zero. But it's not true if a is negative -- if you plug a = -3 into √(a^2), you'll see that it is equal to 3, so the sign has flipped from negative to positive. In general, all we can say is that √(a^2) = |a| is always true.

So √(x-2)^2 = |x-2|, and our equation here becomes:

x + |x - 2| = 2

We can solve this using cases:

- if the thing inside the absolute value, x-2, is positive or zero, the absolute value will do nothing, so we can erase it and solve. Then we get

x + x - 2 = 2
2x = 4
x = 2

So when x - 2 > 0, there is only one solution, x = 2.

- when x-2 is negative, though, the absolute value will flip it's sign, so it will become 2-x. So when x < 2, our equation becomes

x + 2 - x = 2
2 = 2

and the equation is always true for every value of x < 2.

Combining our solutions from the two cases, the equation will be true whenever x < 2. If that were an answer choice, it would be the right answer, but we don't find that among the choices. But if we know for certain that x < 2, then it's clearly true that x < 3, so that's the right answer here. We can't be certain any of the other answer choices are correct, because some potential values of x lie outside the intervals in every other choice.

Nice question!

Why can't E be the Answer ?
GMAT Tutor
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Q51  V47
anmolsd1995 wrote:
Why can't E be the Answer ?

Because the equation in the question in the question is true for, say, x = -1, so it does not need to be true that x is between 0 and 2.
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