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

First, let \( y = |x - 1| \). The equation becomes:

\[ y^2 - 5y + 6 = 0 \]

This is a quadratic equation that can be solved using the factorization method. It factors as:

\[ (y - 2)(y - 3) = 0 \]

So, the solutions for \( y \) are \( y = 2 \) and \( y = 3 \).

Re-substitute \( y = |x - 1| \):

1. For \( y = 2 \), we have:

\[ |x - 1| = 2 \Rightarrow x - 1 = 2 \quad \text{or} \quad x - 1 = -2 \]

\[ x = 3 \quad \text{or} \quad x = -1 \]

2. For \( y = 3 \), we have:

\[ |x - 1| = 3 \Rightarrow x - 1 = 3 \quad \text{or} \quad x - 1 = -3 \]

\[ x = 4 \quad \text{or} \quad x = -2 \]

The roots of the original equation \( |x - 1|^2 - 5|x - 1| + 6 = 0 \) are \( x = 3, -1, 4, -2 \).

The sum of these roots is:

\[ 3 + (-1) + 4 + (-2) = 4 \]

The calculated sum of the roots is 4, which falls within the expected range [4, 4].

Thus, the sum of the roots is 4.