The set of vlaue of `k (k in R)` for which the equation `x ^(2) -4 |x|+3 -|k-1|=0` will have exactly four real roots, is:

Options

A. `(-2,4)`
B. `(-4,4)`
C. `(-4,2)`
D. `(-1,0)`

Answer

To solve the equation \( x^2 - 4|x| + 3 - |k - 1| = 0 \) for the set of values of \( k \) such that the equation has exactly four real roots, we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 - 4|x| + 3 - |k - 1| = 0 \]
We can rearrange this to:
\[ x^2 - 4|x| + 3 = |k - 1| \]

### Step 2: Analyze the left-hand side
Let \( f(x) = x^2 - 4|x| + 3 \). We need to analyze the behavior of this function.

### Step 3: Consider cases for \( |x| \)
Since \( |x| \) behaves differently for positive and negative values of \( x \), we will consider two cases:

1. **Case 1:** \( x \geq 0 \) (then \( |x| = x \)):
\[ f(x) = x^2 - 4x + 3 \]
This is a quadratic equation that opens upwards.

2. **Case 2:** \( x < 0 \) (then \( |x| = -x \)):
\[ f(x) = x^2 + 4x + 3 \]
This is also a quadratic equation that opens upwards.

### Step 4: Find the roots of \( f(x) \)
For both cases, we can find the roots of the respective quadratic equations.

1. **For \( x \geq 0 \):**
\[ f(x) = x^2 - 4x + 3 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} \]
This gives us:
\[ x = 3 \quad \text{and} \quad x = 1 \]

2. **For \( x < 0 \):**
\[ f(x) = x^2 + 4x + 3 = 0 \]
Again using the quadratic formula:
\[ x = \frac{-4 \pm \sqrt{16 - 12}}{2} = \frac{-4 \pm 2}{2} \]
This gives us:
\[ x = -1 \quad \text{and} \quad x = -3 \]

### Step 5: Determine the minimum value of \( f(x) \)
The minimum value of \( f(x) \) occurs at the vertex of the parabolas. For \( f(x) = x^2 - 4x + 3 \), the vertex \( x = 2 \):
\[ f(2) = 2^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \]
For \( f(x) = x^2 + 4x + 3 \), the vertex \( x = -2 \):
\[ f(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1 \]
Thus, the minimum value of \( f(x) \) is \(-1\).

### Step 6: Set conditions for \( |k - 1| \)
For the equation to have exactly four real roots, the value of \( |k - 1| \) must be between the minimum value of \( f(x) \) and the maximum value of \( f(x) \) at the roots:
\[ -1 < |k - 1| < 3 \]

### Step 7: Solve the inequalities
1. From \( |k - 1| > -1 \) (always true since absolute values are non-negative).
2. From \( |k - 1| < 3 \):
\[ -3 < k - 1 < 3 \]
This simplifies to:
\[ -2 < k < 4 \]

### Conclusion
The set of values of \( k \) for which the equation has exactly four real roots is:
\[ k \in (-2, 4) \]