Q26 / 75 · JEE Main 2026 (21 January Shift 1) · Mathematics
The sum of all the roots of the equation $(x-1)^{2}-5|x-1|+6=0$, is :
- A1
- B5
- C3
- Correct answerD4
Solution
Let $t = |x-1|$ where $t \geq 0$.
Equation becomes $t^2 - 5t + 6 = 0$.
$(t-2)(t-3) = 0$ gives $t = 2$ or $t = 3$.
For $|x-1| = 2$: $x = 3$ or $x = -1$.
For $|x-1| = 3$: $x = 4$ or $x = -2$.
Sum of roots $= 3 + (-1) + 4 + (-2) = 4$.
Equation becomes $t^2 - 5t + 6 = 0$.
$(t-2)(t-3) = 0$ gives $t = 2$ or $t = 3$.
For $|x-1| = 2$: $x = 3$ or $x = -1$.
For $|x-1| = 3$: $x = 4$ or $x = -2$.
Sum of roots $= 3 + (-1) + 4 + (-2) = 4$.
Hence, the correct answer is option: 'D'.