Question

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

Hint:

For equations with absolute values, substitute the absolute value expression with a variable and solve the resulting equation.

Answer 1

Correct Answer: 4

Step 1: Let \( |x - 1| = t \).
Substitute \( |x - 1| = t \) into the equation: \[ t^2 - 5t + 6 = 0 \] Solve this quadratic equation: \[ (t - 3)(t - 2) = 0 \] So, \( t = 2 \) or \( t = 3 \). Step 2: Solve for \( x \).
For \( t = 2 \), \( |x - 1| = 2 \) gives \( x = 3 \) or \( x = -1 \). For \( t = 3 \), \( |x - 1| = 3 \) gives \( x = 4 \) or \( x = -2 \). Step 3: Calculate the sum of the roots.
The roots are \( 3, -1, 4, -2 \), so the sum is: \[ 3 + (-1) + 4 + (-2) = 4 \]