The sum of roots of the equation `x^2−∣2x−3∣−4=0` is

Options

A. `2`
B. `-2`
C. `2^(1/2)`
D. `-2^(1/2)`

Answer

To solve the equation \( x^2 - |2x - 3| - 4 = 0 \) and find the sum of its roots, we will consider two cases based on the definition of the absolute value.

### Step 1: Define the cases for the absolute value
The expression \( |2x - 3| \) can be defined in two cases:
1. **Case 1:** \( 2x - 3 \geq 0 \) (i.e., \( x \geq \frac{3}{2} \))
- Here, \( |2x - 3| = 2x - 3 \).
2. **Case 2:** \( 2x - 3 < 0 \) (i.e., \( x < \frac{3}{2} \))
- Here, \( |2x - 3| = -(2x - 3) = 3 - 2x \).

### Step 2: Solve Case 1
For \( x \geq \frac{3}{2} \):
\[ x^2 - (2x - 3) - 4 = 0 \]
This simplifies to:
\[ x^2 - 2x + 3 - 4 = 0 \implies x^2 - 2x - 1 = 0 \]

### Step 3: Find the roots for Case 1
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 1, b = -2, c = -1 \).
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \]
Thus, the roots are \( 1 + \sqrt{2} \) and \( 1 - \sqrt{2} \).

### Step 4: Check which root is valid for Case 1
- \( 1 + \sqrt{2} \) is greater than \( \frac{3}{2} \) (valid).
- \( 1 - \sqrt{2} \) is less than \( \frac{3}{2} \) (not valid).

### Step 5: Solve Case 2
For \( x < \frac{3}{2} \):
\[ x^2 - (3 - 2x) - 4 = 0 \]
This simplifies to:
\[ x^2 + 2x - 3 - 4 = 0 \implies x^2 + 2x - 7 = 0 \]

### Step 6: Find the roots for Case 2
Using the quadratic formula again:
- Here, \( a = 1, b = 2, c = -7 \).
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 28}}{2} = \frac{-2 \pm \sqrt{32}}{2} = \frac{-2 \pm 4\sqrt{2}}{2} = -1 \pm 2\sqrt{2} \]
Thus, the roots are \( -1 + 2\sqrt{2} \) and \( -1 - 2\sqrt{2} \).

### Step 7: Check which root is valid for Case 2
- \( -1 + 2\sqrt{2} \) is approximately \( 1.828 \) (valid).
- \( -1 - 2\sqrt{2} \) is negative (valid).

### Step 8: Calculate the sum of the valid roots
The valid roots are:
- From Case 1: \( 1 + \sqrt{2} \)
- From Case 2: \( -1 + 2\sqrt{2} \)

Now, we calculate the sum:
\[ \text{Sum} = (1 + \sqrt{2}) + (-1 + 2\sqrt{2}) = 1 + \sqrt{2} - 1 + 2\sqrt{2} = 3\sqrt{2} \]

### Final Answer
The sum of the roots of the equation \( x^2 - |2x - 3| - 4 = 0 \) is \( 3\sqrt{2} \).