If sqrt(x^2 + (sqrt(x^2 + sqrt(x^2 + ...))), where the given expressio

Question

If  \sqrt{x^2+{\sqrt{x^2+\sqrt{x^2+\sqrt{x^2+...}}}}}=9 , where the given expression extends to an infinite number of roots, then what is the product of all the possible values of  x ?

A. -72
B. 3√2
C. 6√2
D. 36
E. 72

Are You Up For the Challenge: 700 Level Questions

GMATWhizTeam · Accepted Answer

Solution: 
 •  \sqrt{x^2 +\sqrt{x^2 +\sqrt{x^2 + \sqrt{x^2 + ...}}}} =9 …(i)
• Squaring both sides, we get
•  x^2 + \sqrt{x^2 + \sqrt{x^2 + \sqrt{x^2 + \sqrt{x^2+...}}}} =81  
On substituting the value from equation 1, we get
 •  x^2 + 9 = 81 
•  x^2 = 72 
•  x =6\sqrt{2}\ or\ x= - 6\sqrt{2}  
Their product =  6\sqrt{2}* -6\sqrt{2}= -72 
Hence, the correct answer is  Option A.

Bunuel · Answer

madzaka · Answer

we can write it as
 sqrt(x^2+9)=9

then we power it to the 2nd

x^2+9=81

and result
 x=+/- 6*sqrt(2)

Ans  C

TestPrepUnlimited · Answer

This is actually a really fun brainteaser!

Since we know the entire left side is equal to 9, notice part of what is under the square root is equal to the original left side. We can replace any of the sqrt(....) parts with 9, so let us replace the second square root with 9 to get:

\sqrt{x^2 + 9} = 9 
 x^2 + 9 = 81 
 x^2 = 72

The question is asking for the  product of all roots , we have a quadratic equation so likely we get two solutions.
Note one of the roots is positive and the other is negative, and both have a magnitude of  \sqrt{72} . If we multiply those two roots we get  -72 .

Ans: A

sambitspm · Answer

See the attachment
Answer A

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem Series ↧↧↧

https://www.youtube.com/watch?v=vCMud0BE1H0

Given that  \sqrt{x^2+{\sqrt{x^2+\sqrt{x^2+\sqrt{x^2+...}}}}}=9  and we need to find product of all the possible values of x

Now the expression inside the square roots extends till infinity and if we note then the terms are repeating.
So, we can approximate the problem by writing terms after the first  \sqrt{x^2}  = 9

=>  \sqrt{x^2 + 9}=9

Squaring both the sides
 x^2 + 9 = 9^2 
 x^2  = 81 - 9 = 72
=> x = ±  \sqrt{72} 
=> x = +  \sqrt{72}  or x = -  \sqrt{72}

=> Product of the two values of x = +  \sqrt{72}  * -  \sqrt{72}  = -72

So,  Answer will be A 
Hope it helps!

Watch the following video to MASTER Roots

https://www.youtube.com/watch?v=5oAec-Y6kOQ

If sqrt(x^2 + (sqrt(x^2 + sqrt(x^2 + ...))), where the given expressio

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If sqrt(x^2 + (sqrt(x^2 + sqrt(x^2 + ...))), where the given expressio

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards