What is the sum of all possible value of x for which 4^x – 2^{x + 3} –

Question

What is the sum of all possible value of x for which  4^x – 2^{x + 3} – 2^{x + 2} + 32 = 0?

(A) 5

(B) 9

(C) 11

(D) 14

(E) 20

prathyushaR · Accepted Answer

Lets assume  2^x = n   
so: 4^x=n^2 ; 2^{x + 3}=n*2^3 ; 2^{x + 2}=n*2^2 
eqation becomes: 
 n^2-8n-4n+32=0: n^2-12n+32=0  
 :(n-8)(n-4)=0 :n=4 or 8 : 2^x =    4 or 8 : x=2 or 3 
the sum comes out as 5
ANSWER :A

ArunSharma12 · Answer

4^x – 2^{x + 3} – 2^{x + 2} + 32 = 0 
 2^{2x} – 8*2^{x} – 4*2^{x} + 32 = 0 
 2^{2x} – 12*2^{x} + 32 = 0 
let y =  2^x 
 y^{2} – 12*y + 32 = 0 
solving this equation, we'll get y = 8,4
 2^x=2^3 ; x =3
 2^x=2^2 ; x =2
sum of all possible values of x = 5
Ans: A

GMATWhizTeam · Answer

Solution: 
 o  4^x-2^{x+3}-2^{x+2} + 32 = 0 
o  2^{2x} – 2^{x+3} – 2^{x+2} + 2^5 = 0 
o  2^5 = 2^{x+3} + 2^{x+2} – 2^{2x} 
o  2^5 = 2^{x}  
o  2^5 = 2^x  
o  2^5 = 2^x  
o  2^5 = 2^x  
o  2^5 = 2^{x+2}   
Now, LHS is positive, the RHS will also be positive
 o We know that,  2^{x+2}  is always positive
 o  3 – 2^{x-2}  must be positive
o  2^{x-2} < 3 
 o The only possible values of x are  2  and  3   
o Thus, the required Sum is  2 + 3 = 5  .
Hence, the correct answer is  Option A.

Fdambro294 · Answer

It’s important to use the many exponent rules both “backwards and forwards”

For each term, “back out” the multiplication of common bases and different exponents:

(4)^x = (2^2)^x = (2)^2x = (2)^x * (2)^x

- (2)^x + 3 = -(2)^x * (2)^3

- (2)^x + 2 = -(2)^x * (2)^2

And for the last term, we want to rearrange 32 to “match” the other terms we have rearranged:

32 = (2)^5 = (2)^2 * (2)^3

Putting all the rearranged terms together and setting them equal to 0, you can see the “hidden quadratic” (essentially the equations can be written as the product of 2 binomials) more clearly:

(2)^x * (2)^x - (2)^2 * (2)^x - (2)^3 * (2)^x + (2)^2 * (2)^3 = 0

(2^x - 2^2) * (2^x - 2^3) = 0

Rearranging the terms as the product of two factors that must equal zero, we can use the zero product rule/concept in that either one of the binomials must equal = 0

(2)^x = (2)^2

Or

(2)^x = (2)^3

Therefore

X = 2

Or

X = 3

Sum: 2 + 3 = 5

Answer (A)

Trickier question than the percentage correct suggests….

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the sum of all possible value of x for which 4^x – 2^{x + 3} –

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

What is the sum of all possible value of x for which 4^x – 2^{x + 3} –

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