Consider the equation |x^2-2x- 3|=m .m belongs to R .If the given equation has four solutions, then `m in (0,oo)` b. `m in (-1,3)` c. `m in (0,4)` d. None of these

Options

A. `m in (0,oo)`
B. `m in (-1,3)`
C. `m in (0,4)`
D. None of these

Answer

To solve the equation |x² - 2x - 3| = m, we need to analyze the quadratic function f(x) = x² - 2x - 3 and determine the conditions under which the absolute value equation has four solutions.

### Step 1: Find the roots of the quadratic equation
First, we need to find the roots of the quadratic equation f(x) = x² - 2x - 3 = 0.

Using the factorization method:
- We can rewrite the equation as:
\[ x² - 2x - 3 = (x - 3)(x + 1) = 0 \]
- The roots are:
\[ x = 3 \quad \text{and} \quad x = -1 \]

### Step 2: Analyze the graph of the quadratic function
The graph of the quadratic function f(x) = x² - 2x - 3 is a parabola that opens upwards. The vertex of the parabola can be found using the vertex formula \( x = -\frac{b}{2a} \):
- Here, \( a = 1 \) and \( b = -2 \):
\[ x = -\frac{-2}{2 \cdot 1} = 1 \]
- Now, substituting \( x = 1 \) back into f(x) to find the y-coordinate of the vertex:
\[ f(1) = 1² - 2(1) - 3 = 1 - 2 - 3 = -4 \]
- Thus, the vertex is at the point (1, -4).

### Step 3: Determine the behavior of the absolute value function
Since we are considering the absolute value |f(x)|, the graph will reflect any negative portions of the parabola above the x-axis:
- The points where f(x) = 0 are x = -1 and x = 3.
- The vertex (1, -4) is below the x-axis, so the absolute value will create a "V" shape from this vertex.

### Step 4: Find the conditions for four intersections with the line y = m
For the equation |f(x)| = m to have four solutions, the horizontal line y = m must intersect the graph of |f(x)| at four distinct points:
- This occurs when the line y = m is above the vertex (1, -4) and below the maximum point of the absolute value function.
- The maximum point occurs at the vertex of the original parabola, which is at y = -4. Therefore, for the line y = m to intersect the graph four times, m must be greater than 0 (to be above the x-axis) and less than 4 (to be below the maximum point of the absolute value function).

### Conclusion
Thus, the condition for m is:
\[ m \in (0, 4) \]

From the given options, the correct answer is:
**c. m in (0, 4)**