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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 :
\((x^2−25)^2\) = \([(x-5)(x+5)]^2\) = \((x-5)^2(x+5)^2\)

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
\((x-5)^2 [(x+5)^2 - 1] = 0\)
Simplifying we will get :
\((x-5)^2 [x^2+10x+24] = 0\)
solve this quadratic to factor form we will get:
\((x-5)^2 [(x+7)(x+3)] = 0\)

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

Hence IMO C.
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Bunuel
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


 


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Bunuel
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


 


This question was provided by GMAT Club
for the Around the World in 80 Questions

Win over $20,000 in prizes: Courses, Tests & more

 


\((x^2 - 25)^2 = (x + 5)^2 (x - 5)^2 \)
\(x^2 - 10x + 25 = (x - 5)^2 \)
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
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Bunuel
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


 


This question was provided by GMAT Club
for the Around the World in 80 Questions

Win over $20,000 in prizes: Courses, Tests & more

 


(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
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