Around the World in 80 Questions (Day 2): What is the sum of all value

Question

What is the sum of all value of x such that  (x^2 - 25)^2 = x^2 - 10x + 25  ?

A. -10
B. -5
C. 0
D. 1
E. 5

Kinshook · Answer

Asked: What is the sum of all value of x such that  (x^2 - 25)^2 = x^2 - 10x + 25  ?

(x^2 - 25)^2 =(x-5)^2(x+5)^2= x^2 - 10x + 25 = (x-5)^2  
 (x-5)^2(x+5)^2- (x-5)^2=0  
 (x-5)^2{(x+5)^2- 1}=0  
 (x-5)^2(x+4)(x+6)=0  
x = {5,-4,-6}
Sum of all values of x = 5-4-6=-5

IMO B

missionmba2025 · Answer

^2 = (x-5)^2

(x+5)^2(x-5)^2 - (x-5)^2 = 0

(x-5)^2   = 0
 
 1) (x - 5) ^ 2 = 0 ; x = 5

2) (x+5)^ 2 = 1

Taking square root on both sides

|x+5| = 1

x = -4 or x = -6

Sum = -6 + -4 + 5 = -10 + 5 = -5

IMO B

IshanSaini · Answer

What is the sum of all value of x such that (x^2−25)^2=x^2−10x+25

Start with simplifying the left side :
 (x^2−25)^2  =   ^2  =  (x-5)^2(x+5)^2

Now move to the right side of the equation :
 x^2−10x+25  Solving this we will get Discriminant = 0 and thus -b/2a = 5 so  (x-5)^2  when written in simplified factor form.

CAREFUL NOW !
dont blindly cancel them.

INSTEAD

write it as 
 (x-5)^2 (x+5)^2 - (x-5)^2 = 0 
or 
 (x-5)^2   = 0 
Simplifying we will get :
 (x-5)^2   = 0 
solve this quadratic to factor form we will get:
 (x-5)^2   = 0

So the values of x are +5 +5 -7 - 3 = 0

Hence IMO C.

TargetKellogg2024 · Answer

cledgard · Answer

(x^2 - 25)^2 = (x + 5)^2 (x - 5)^2   
 x^2 - 10x + 25 = (x - 5)^2  
Therefore 
  (x + 5)^2 (x - 5)^2 = (x - 5)^2  
We can divide both sides by   (x - 5)^2   , and   one value of x is 5 ,  because that would make both sides 0.
We still have   (x + 5)^2 = 1   
Taking the square root of both sides we get   |x + 5| = 1   so   x = -4  , and   x = -6 . 
If we add the 3 values of x, it becomes   5 - 4 - 6 = -5  
Answer B

sujoykrdatta · Answer

(x^2 - 25)^2 = x^2 - 10x + 25
=> (x^2 - 25)^2 = (x - 5)^2
=> (x^2 - 25)^2 - (x - 5)^2 = 0
=> (x^2 - 25 - x + 5)(x^2 - 25 + x - 5) = 0 ..... using a^2 - b^2 = (a + b)(a - b) 
=> (x^2 - x - 20)(x^2 + x -30) = 0
=> (x - 5)(x + 4)(x + 6)(x - 5) = 0
=> x = -6, -4 or 5
Sum = -5

Answer B

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.

Around the World in 80 Questions (Day 2): What is the sum of all value

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

Around the World in 80 Questions (Day 2): What is the sum of all value

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