JEE Mathematics 2026 Question with Solution

Given: The equation is $(x-1)^2 - 5|x-1| + 6 = 0$. We have to find the sum of all its roots.

Find: The sum of all values of $x$ satisfying the equation.

Use the substitution $y = |x-1|$. Since $(x-1)^2 = |x-1|^2$, the equation becomes

$$y^2 - 5y + 6 = 0$$

Factoring,

$$(y-2)(y-3) = 0$$

So,

$$y = 2 \quad \text{or} \quad y = 3$$

Now substitute back:

$$|x-1| = 2$$

which gives

$$x-1 = 2 \Rightarrow x = 3$$

and

$$x-1 = -2 \Rightarrow x = -1$$

Also,

$$|x-1| = 3$$

which gives

$$x-1 = 3 \Rightarrow x = 4$$

and

$$x-1 = -3 \Rightarrow x = -2$$

Thus the roots are $-2, -1, 3, 4$. Their sum is

$$(-2)+(-1)+3+4 = 4$$

Therefore, the sum of all the roots is $4$, so the correct option is C.

The key observation is that the equation contains both $|x-1|$ and $(x-1)^2$. Writing $y = |x-1|$ converts it into a quadratic in one variable.

Then solve

$$y^2 - 5y + 6 = 0$$

by factorization to get $y=2$ and $y=3$.

For each value of $y$, solve the corresponding absolute value equation:

• From $|x-1|=2$, the roots are $x=3$ and $x=-1$.
• From $|x-1|=3$, the roots are $x=4$ and $x=-2$.

Adding all roots,

$$3 + (-1) + 4 + (-2) = 4$$

Hence, the required sum is $4$.