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Let k be a real number. If the equation x^2 − 2x = k|x| has exactly three different real roots, find the value range of k.

We are given the equation:

\[
x^2 - 2x = k |x|
\]

where \( k \) is a real number, and the equation has exactly three different real roots. We are tasked with finding the value range of \( k \).

### Step 1: Analyze the Equation
We need to solve the equation by considering two cases due to the presence of the absolute value term \( |x| \):

- **Case 1:** \( x \geq 0 \) implies \( |x| = x \).
- **Case 2:** \( x < 0 \) implies \( |x| = -x \).

#### Case 1: \( x \geq 0 \)
In this case, the equation becomes:

\[
x^2 - 2x = kx
\]

Rearranging terms:

\[
x^2 - (2 + k)x = 0
\]

Factoring out \( x \):

\[
x(x - (2 + k)) = 0
\]

Thus, the solutions for \( x \geq 0 \) are:

\[
x = 0 \quad \text{or} \quad x = 2 + k
\]

#### Case 2: \( x < 0 \)
In this case, the equation becomes:

\[
x^2 - 2x = -kx
\]

Rearranging terms:

\[
x^2 - (2 - k)x = 0
\]

Factoring out \( x \):

\[
x(x - (2 - k)) = 0
\]

Since \( x \neq 0 \) in this case (we are only considering \( x < 0 \)), the solution is:

\[
x = 2 - k
\]

### Step 2: Roots Analysis
Now, we have the potential roots:

- From **Case 1**, \( x = 0 \) and \( x = 2 + k \).
- From **Case 2**, \( x = 2 - k \).

We are given that there are exactly three different real roots. This means that:
- \( 2 + k \) must be positive (since it belongs to \( x \geq 0 \)),
- \( 2 - k \) must be negative (since it belongs to \( x < 0 \)),
- And \( 0 \) is a root.

Thus, the three roots are \( 0 \), \( 2 + k \), and \( 2 - k \).

For the roots to be distinct:
- \( 2 + k > 0 \), implying \( k > -2 \),
- \( 2 - k < 0 \), implying \( k > 2 \).

However, \( k > 2 \) already satisfies both conditions.

### Step 3: Conclusion
The range of \( k \) is:

\[
k > 2
\]

Would you like more details or explanations? Here are some related questions for further exploration:

1. How do we determine the behavior of absolute value functions in equations?
2. Can we verify that exactly three real roots occur by testing specific values of \( k \)?
3. What is the significance of root multiplicity in polynomial equations?
4. How does the inequality \( k > 2 \) ensure distinct roots?
5. How do factorization techniques help solve absolute value equations?

**Tip:** When solving equations with absolute values, always consider splitting the equation into different cases based on the definition of the absolute value.

Discover how to determine the value range of k in the equation x^2 - 2x = k|x|, where the equation has exactly three different real roots. This solution explores the properties of absolute value functions, quadratic equations, and factorization techniques. Suitable for students in Grades 11-12.

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