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The sum of all the real roots of the equation |x – 2| 2 + |x – 2| - 2 = 0 is . . . . . . . . . . . . . . . . . . . . . . .

The sum of all the real roots of the equation |x – 2|2 + |x – 2| - 2 = 0 is . . . . . . . . . . . . . . . . . . . . . . .

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
7 years ago
|x – 2|2 + |x – 2| - 2 = 0
Case 1. x ≥ 2
⇒ (x – 2)2 + (x – 2) – 2 = 0
⇒ x2 – 3x = 0 ⇒ x(x – 3) = 0
⇒ x = 0, 3(0 is rejected as x ≥ 2)
⇒ x = 3 . . . . . . . . . . . . . . (1)
Case 2. X < 2
{ - (x – 2)}2 – (x – 2) – 2 = 0
⇒ x2 + 4 – 4x – x = 0 ⇒ (x – 1) (x – 4) = 0
⇒ x = 1, 4 (4 is rejected as x < 2)
⇒ x = 1 . . . . . . . . . . . . . (2)
Therefore, the sum of the roots is 3 + 1 = 4.

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