# Lesson: Equations with Square Roots

## Comment on Equations with Square Roots

### Thanks for very great

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? ### Extraneous roots aren't

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

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 ### Good question. Also a very

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

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

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? 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

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. 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.

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!

Hi Brent!

I'm still not clear on what an extraneous root is. Could you please explain this a bit more? ### An extraneous root is a

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, if √25 cannot be -5

Hi Brent, if √25 cannot be -5, then in other DS questions, why do we call a statement insufficient when we get answer as root of a number, saying that it could be + or -? As far as i know, root of 25 is +5 and -5. ### Hi Kashaf,

Hi Kashaf,

Your're correct to say that every positive number (other than 1) has two square roots.
However the square root NOTATION tells us to take the only 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.

I believe you're thinking of DS questions, where a statement might say that x² = 25
In this case, x = √25 or x = -√25
When we evaluate, we get: x = 5 or x = -5

Does that help?

Cheers,
Brent

### Brent, great explanation

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. Very nice!

### Hi Brent,

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 ### Good question!

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,

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 ### You bet.

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,

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.
Thank you! 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

### At 04:38 are we allowed to

At 04:38 are we allowed to expand (a + b)^2 as (a + b) * (a + b), instead of a^2 + b^2 + 2ab ?? ### It's acceptable to write (x +

It's acceptable to write (x + 2)² as (x + 2)(x + 2).
However, if you do that, I think you'll find that your solution will come to a halt if you don't expand and simplify (x + 2)(x + 2) to get: x² + 4x + 4

Cheers,
Brent