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Answer:
0
Step-by-step explanation:
For convenience, let √x = a.
⇒ | a - 2 | + a( a - 4 ) + 2 = 0
⇒ | a - 2 | + a^2 - 4a + 2 = 0
Now, two cases arise:
Case 1:
⇒ ( a - 2 ) + a^2 - 4a + 2 = 0
⇒ a - 2 + a^2 - 4a + 2 = 0
⇒ a^2 - 4a + a - 2 + 2 = 0
⇒ a^2 - 3a = 0
⇒ a( a - 3 ) = 0
⇒ a = 0 or a = 3
This means, either √x = 0 or √x = 3. As given, x ≠ 0, therefore, √x = 3 ⇒ x = 9.
Case 2:
⇒ - ( a - 2 ) + a^2 - 4a + 2 = 0
⇒ - a + 2 + a^2 - 4a + 2 = 0
⇒ a^2 - 4a - a + 2 + 2 = 0
⇒ a^2 - 4a - a + 4 = 0
⇒ a( a - 4 ) - ( a - 4 ) = 0
⇒ ( a - 4 )( a - 1 ) = 0
⇒ a = 4 or a = 1
Thus, √x = 4 or √x = 1 ⇒ x = 16 or x = 1.
But,
| √x - 2 | + √x( √x - 4 ) + 2 = 4 ≠ 0, for x = 16.
Therefore, x = 1
Hence,
Solutions for the given equation are 9 and 1.
Thus, k = 2 ⇒ k - 2 = 2 - 2 = 0 ⇒ k - 2 = 0
Hence the required value of k - 2 is 0.
Answer:
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