Lesson: Equations with Square Roots

Comment on Equations with Square Roots

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

Hi Brent!

I'm still not clear on what an extraneous root is. Could you please explain this a bit more?
gmat-admin's picture

An extraneous root is a solution to an equation you got after performing all steps correctly, YET the solution does not satisfy the original equation.

Sometimes these extraneous roots occur because SQUARING numbers will turn BOTH positive values and negative values into POSITIVE values.

Here's a rudimentary example: √(x + 3) = -5
Square both sides to get: [√(x + 3)]² = (-5)²
Simplify: x + 3 = 25
Solve: x = 22

When we plug x = 22 back into the original equation,
We get: √(22 + 3) = -5
Simplify: √25 = -5
Hmm, no good.

So, x = 22 is an extraneous root.

It turns out that the equation √(x + 3) = -5 has no solution.
This should make sense, since √(some value) CANNOT evaluate to be a NEGATIVE number (like -5)

Does that help?

Cheers,
Brent

hi Brent, I understand the concept behind extraneous root. but I do no understand in GMAT question context. can you please provide an example from the OG?
gmat-admin's picture

Great question!!

This is a concept/situation that the test-makers CAN/COULD exploit, but I cannot readily find an example of such a question.

Here's an example of how an official question might look:

How many roots does the equation √(x² - 6) = √x have?
A) zero
B) one
C) two
D) three
E) four

When we use the usual techniques to solve √(x² - 6) = √x, we get two solutions: x = 3 and x = -2
However, since x = -2 is an extraneous root, there is only one solution (x = 3)

Answer: B

Cheers,
Brent

Brent, great explanation again! https://gmatclub.com/forum/if-y-root-3y-4-then-the-product-of-all-possible-solution-94527.html
The guys on the forum are struggling with this question wondering why the -ve root is not considered.
I easily answered this because of your crisp explanation on extraneous roots.
gmat-admin's picture

Very nice!

Hi Brent,

if I get an equation such as the following do I need to check on both sides of the equation if the root is negative or is it sufficient to check either one?

√(2z^2 - 7z + 49) = √(z^2 + 7z)

Thank you!
Pia
gmat-admin's picture

Good question!
If there are square roots on both sides of the original equation, you'll have to check both sides.
Cheers,
Brent

Hi Brent,
Can you please explain how to solve this question?
https://gmatclub.com/forum/if-root-3-2x-root-2x-1-then-4x-135539.html
Thanks,
Rena
gmat-admin's picture

You bet.
Here's my full solution: https://gmatclub.com/forum/if-root-3-2x-root-2x-1-then-4x-135539-20.html...

Cheers,
Brent

Hi Brent,

Can you please help me with the solution? While the solution explained is great but I cannot understand the logic behind the 1st statement.
Question link - https://gmatclub.com/forum/if-n-is-positive-is-root-n-166513.html
Thank you!
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-n-is-positive-is-root-n-166513.html

Hi Harleen,

Have you seen my solution here: https://gmatclub.com/forum/if-n-is-positive-is-root-n-166513.html#p1725651 ?

If so, can you tell me where I lost you?

Cheers,
Brent

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