Question

The sum of roots of the equation \[ |x - 1|^2 - 5|x - 1| + 6 = 0 \] is:

Hint:

For equations involving absolute values: \begin{itemize} \item Substitute the absolute value expression to simplify \item Solve the resulting algebraic equation \item Don’t forget to find all corresponding values of the variable \end{itemize}

Answer 1

Correct Answer: 4

Step 1: Let \[ |x - 1| = t \] Then the given equation becomes: \[ t^2 - 5t + 6 = 0 \]
Step 2: Factorize: \[ (t - 2)(t - 3) = 0 \] \[ \Rightarrow t = 2 \text{or} t = 3 \]
Step 3: Replace \( t = |x - 1| \): \[ |x - 1| = 2 \Rightarrow x = 3, -1 \] \[ |x - 1| = 3 \Rightarrow x = 4, -2 \]
Step 4: Sum of all roots: \[ 3 + (-1) + 4 + (-2) = 4 \]