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.

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