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Question 18 : In how many ways can a pair of integers (x , a) be chosen such that x^{2} − 2 | x | + | a - 2 | = 0 ?

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|x|^{2} – 2|x| + |a – 2| = 0

|x|^{2} – 2|x| + 1 = 0 is the square of a quadratic number

In the above equation the value of constant cannot be more than 1

So |a – 2| = 0 or = 1

|x|^{2} – 2|x| = 0

|x|^{2} = 2|x|

x = 0 or 2 or -2

For all these possibilities value of a = 2

|x|^{2} – 2|x| + 1 = 0

(|x| - 1)^{2} = 0

|x| = 1

So, x = 1 or x = -1

Then |a – 2| = 1, a = 3 or a = 1

Four combinations of (x,a) are possible already we have 3

Totally 7 pairs

The question is **"In how many ways can a pair of integers (x , a) be chosen such that x ^{2} − 2 | x | + | a - 2 | = 0 ?" **

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