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What is the sum of the squares of all real values of $x$ for which $|2-|x| |=1$?

 Jan 25, 2021
 #1
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What did you get for an answer?

 Jan 25, 2021
 #2
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I got 20. I got this because 1, -1, 3, and -3 work. 1² is 1, -1² is 1, 3² is 9, and -3² is 9. 1+1+9+9=20.

Guest Jan 25, 2021
 #3
avatar+116126 
+1

We can write  l x l  as   sqrt (x^2)

 

So....this implies that....

 

sqrt  [ ( (2 -  sqrt (x^2) )^2  ]     =  1          square both sides

 

( (2 - sqrt (x^2))^2     =   1         simplify

 

4 - 4sqrt (x^2)  +  x^2    = 1

 

x^2 +  3  =  4sqrt (x^2)            square both sides again

 

x^4  + 6x^2   + 9  =  16x^2

 

x^4  - 10x^2 + 9   =   0       factor

 

(x^2 - 9) ( x^2 - 1)   =  0

 

x^2  - 9   =  0            x^2  -1  = 0

x = 3 , -3                   x =  -1, 1           which  are the values  found  by the guest   !!!

 

 

cool cool cool

 Jan 25, 2021

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