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.