Lesson: Equations with Square Roots

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Thanks for very great explanation!
So if we get a question in GMAT that contains extraneous root, we take it as positive, am i correct?
gmat-admin's picture

Extraneous roots aren't necessarily negative. So, we always have to check for extraneous roots.
Consider this example: sqrt(13 - x) = 1 - x
Square both sides to get: 13 - x = (1 - x)^2
Expand: 13 - x = 1 - 2x + x^2
Rearrange: x^2 - x - 12 = 0
Factor: (x + 3)(x - 4) = 0
So, we have two possible solutions: x = -3 and x = 4
Now plug them back in to the original equation to check for EXTRANEOUS roots.

If x = -3, we get: sqrt[13 - (-3)] = 1 - (-3)
Simplify: sqrt(16) = 4
This WORKS, so one solution is x = -3

If x = 4, we get: sqrt(13 - 4) = 1 - 4
Simplify: sqrt(9) = -3
NO GOOD!

In this case, the solution with the NEGATIVE value was the GOOD solution, and the solution with the POSITIVE value was the EXTRANEOUS root.

when (-1)^2 and (1)^2 both is equal to 1 then why cant (1)^1/2 cant equal to both +1 and -1.. can u pls explain
gmat-admin's picture

Good question. Also a very common question.

From the Official Guide:

"A square root of a number n is a number that, when squared, is equal to n. Every positive number n has two square roots, one positive and the other negative, but √n denotes the positive number whose square is n. For example, √9 denotes 3"

So, the square root NOTATION tells us to take the POSITIVE root.

As such, √1 = 1 (and only 1)
Also, -√1 = -1

Hi Brent! Could you please explain this question - https://gmatclub.com/forum/if-4-x-1-2-x-2-then-x-could-be-equal-to-which-of-the-foll-188185.html

Hi Brent! At 1:58 you said a square root of a number can never be negative. I'm confused here. In GMAT what is the Sqrt of 81? 9 or -9? or just 9 alone?
gmat-admin's picture

Hi Mohammad,

Every positive number has two square roots, however the square root NOTATION tells us to take the POSITIVE root.

So, for example, √81 = 9 (only)

From the OFFICIAL GUIDE FOR GMAT REVIEW:
"A square root of a number n is a number that, when squared, is equal to n. Every positive number n has two square roots, one positive and the other negative, but √n denotes the positive number whose square is n. For example, √9 denotes 3"
So, the square root NOTATION tells us to take the POSITIVE root.

https://gmatclub.com/forum/if-4x-12-x-9-which-of-the-following-must-be-true-101732.html

I found the solution x ≥ 7

But the answer says 'A', X > 6
The solution comes x is greater than or equal to 7. So any value less than 7 would not be the solution. Of course If the solution is greater or equal to 7, it would be definitely greater than 6. Why answer would be A? What am I missing? I am confused. Please explain.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-4x-12-x-9-which-of-the-following-must-be-t...

You took the inequality 4x - 12 ≥ x + 9 and correctly simplified it to get x ≥ 7

Now that we know x ≥ 7, we must choose the statement that MUST be true.

C) x > 7
Must this be true?
No.
If x ≥ 7, then x COULD equal 7, in which case, it is NOT true that x > 7
So, answer choice C is not necessarily true.

It all comes down to the words "MUST BE TRUE"

A) x > 6
If x ≥ 7, MUST it be true that x > 6?
Yes.
Answer: A

Here's a similar example:
If we know that Joe is older than 10 years old, which of the following MUST BE TRUE?

A) Joe is older than 11 years old.
MUST this be TRUE?
No.
For example, if Joe is older than 10 years old, then Joe COULD be 11 years old, in which case answer choice A is NOT TRUE

B) Joe is older than 3 years old.
MUST this be TRUE?
YES! If Joe is older than 10 years old, then it MUST BE TRUE that if Joe is older than 3 years old,

Does that help?

Cheers,
Brent

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