Question

The sum of all the roots of the equation \((x-1)^2 - 5|x-1| + 6 = 0\), is:

• A. 5
• B. 3
• C. 4
• D. 1

Hint:

For equations involving \(|f(x)|\) and \(f(x)^2\), the substitution \(y=|f(x)|\) is a very effective technique. It transforms the equation into a standard polynomial, which is easier to solve. Always remember to substitute back and solve for the original variable, considering both positive and negative cases for the absolute value.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given an equation involving an absolute value term. We need to find all the solutions (roots) of this equation and then calculate their sum.
Step 2: Key Formula or Approach:
The equation has the form of a quadratic equation if we make a substitution.
Let \(y = |x-1|\).
Since \((x-1)^2 = |x-1|^2\), the equation can be rewritten in terms of \(y\). After solving for \(y\), we substitute back \(|x-1|\) and solve for \(x\).
Remember that if \(|u|=k\) (for \(k>0\)), then \(u=k\) or \(u=-k\).
Step 3: Detailed Explanation:
The given equation is \((x-1)^2 - 5|x-1| + 6 = 0\). Notice that \((x-1)^2 = |x-1|^2\). Let's substitute \(y = |x-1|\). The equation becomes: \[ y^2 - 5y + 6 = 0 \] This is a simple quadratic equation in \(y\). We can factor it: \[ (y-2)(y-3) = 0 \] This gives two possible values for \(y\): \(y=2\) or \(y=3\). Now, we substitute back \(|x-1|\) for \(y\). Case 1: \(|x-1| = 2\) This implies two possibilities for \(x-1\): \[ x-1 = 2 \implies x = 3 \] \[ x-1 = -2 \implies x = -1 \] So, we have two roots from this case: 3 and -1. Case 2: \(|x-1| = 3\) This implies two possibilities for \(x-1\): \[ x-1 = 3 \implies x = 4 \] \[ x-1 = -3 \implies x = -2 \] So, we have two more roots from this case: 4 and -2. The set of all roots of the equation is \(\{-2, -1, 3, 4\}\). Now, we find the sum of all these roots: \[ \text{Sum} = (-2) + (-1) + 3 + 4 = -3 + 7 = 4 \] Step 4: Final Answer:
The sum of all the roots of the equation is 4.