Number of intergral roots of `|x-1||x^2-2|=2` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To find the number of integral roots of the equation \( |x-1||x^2-2|=2 \), we can break down the problem into manageable steps.

### Step 1: Understand the equation
The equation is given by:
\[ |x-1||x^2-2|=2 \]
This means we need to consider the absolute values of \( x-1 \) and \( x^2-2 \).

### Step 2: Set up cases for absolute values
Since we have absolute values, we can set up cases based on the definitions of absolute values. We will consider two cases:

1. Case 1: \( |x-1| = 1 \) and \( |x^2-2| = 2 \)
2. Case 2: \( |x-1| = 2 \) and \( |x^2-2| = 1 \)

### Step 3: Solve Case 1
For Case 1:
- From \( |x-1| = 1 \):
\[ x - 1 = 1 \quad \text{or} \quad x - 1 = -1 \]
This gives:
\[ x = 2 \quad \text{or} \quad x = 0 \]

- From \( |x^2-2| = 2 \):
\[ x^2 - 2 = 2 \quad \text{or} \quad x^2 - 2 = -2 \]
This gives:
\[ x^2 = 4 \quad \Rightarrow \quad x = 2 \quad \text{or} \quad x = -2 \]
and
\[ x^2 = 0 \quad \Rightarrow \quad x = 0 \]

Now, we take the intersection of the solutions from both parts:
- From \( |x-1| = 1 \): \( x = 0, 2 \)
- From \( |x^2-2| = 2 \): \( x = -2, 0, 2 \)

The common solutions are \( x = 0 \) and \( x = 2 \).

### Step 4: Solve Case 2
For Case 2:
- From \( |x-1| = 2 \):
\[ x - 1 = 2 \quad \text{or} \quad x - 1 = -2 \]
This gives:
\[ x = 3 \quad \text{or} \quad x = -1 \]

- From \( |x^2-2| = 1 \):
\[ x^2 - 2 = 1 \quad \text{or} \quad x^2 - 2 = -1 \]
This gives:
\[ x^2 = 3 \quad \Rightarrow \quad x = \sqrt{3} \quad \text{or} \quad x = -\sqrt{3} \]
and
\[ x^2 = 1 \quad \Rightarrow \quad x = 1 \quad \text{or} \quad x = -1 \]

Now, we take the intersection of the solutions from both parts:
- From \( |x-1| = 2 \): \( x = 3, -1 \)
- From \( |x^2-2| = 1 \): \( x = 1, -1, \sqrt{3}, -\sqrt{3} \)

The common solution is \( x = -1 \).

### Step 5: Combine solutions from both cases
From Case 1, we found \( x = 0, 2 \).
From Case 2, we found \( x = -1 \).

Thus, the integral roots are:
\[ x = -1, 0, 2 \]

### Step 6: Count the number of integral roots
The integral roots we found are \( -1, 0, 2 \). Therefore, the number of integral roots is:
\[ \text{Number of integral roots} = 3 \]

### Final Answer
The number of integral roots of the equation \( |x-1||x^2-2|=2 \) is **3**.

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