Sum of the roots of the equation `x^(2) + |2x - 3| - 4 = 0` is

Options

A. 2
B. `-2`
C. `sqrt(2)`
D. `-sqrt(2)`

Answer

To find the sum of the roots of the equation \( x^2 + |2x - 3| - 4 = 0 \), we will analyze the absolute value and break the problem into two cases.

### Step 1: Analyze the absolute value
The expression \( |2x - 3| \) can be split into two cases based on the value of \( x \):
1. Case 1: \( 2x - 3 \geq 0 \) (i.e., \( x \geq \frac{3}{2} \))
2. Case 2: \( 2x - 3 < 0 \) (i.e., \( x < \frac{3}{2} \))

### Step 2: Solve Case 1 (\( x \geq \frac{3}{2} \))
In this case, we have:
\[ |2x - 3| = 2x - 3 \]
Substituting this into the equation gives:
\[ x^2 + (2x - 3) - 4 = 0 \]
Simplifying this, we get:
\[ x^2 + 2x - 7 = 0 \]

### Step 3: Apply the quadratic formula for Case 1
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, c = -7 \):
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{4 + 28}}{2} \]
\[ x = \frac{-2 \pm \sqrt{32}}{2} \]
\[ x = \frac{-2 \pm 4\sqrt{2}}{2} \]
\[ x = -1 \pm 2\sqrt{2} \]

### Step 4: Check the roots for Case 1
The roots are:
1. \( x_1 = -1 + 2\sqrt{2} \)
2. \( x_2 = -1 - 2\sqrt{2} \)

Since \( 2\sqrt{2} \approx 2.828 \), we find:
- \( x_1 \approx 1.828 \) (valid since \( x_1 \geq \frac{3}{2} \))
- \( x_2 \approx -3.828 \) (not valid since \( x_2 < \frac{3}{2} \))

Thus, only \( x_1 = -1 + 2\sqrt{2} \) is valid in this case.

### Step 5: Solve Case 2 (\( x < \frac{3}{2} \))
In this case, we have:
\[ |2x - 3| = -(2x - 3) = -2x + 3 \]
Substituting this into the equation gives:
\[ x^2 + (-2x + 3) - 4 = 0 \]
Simplifying this, we get:
\[ x^2 - 2x - 1 = 0 \]

### Step 6: Apply the quadratic formula for Case 2
Using the quadratic formula again where \( a = 1, b = -2, c = -1 \):
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]

### Step 7: Check the roots for Case 2
The roots are:
1. \( x_3 = 1 + \sqrt{2} \) (not valid since \( x_3 > \frac{3}{2} \))
2. \( x_4 = 1 - \sqrt{2} \) (valid since \( x_4 < \frac{3}{2} \))

### Step 8: Calculate the sum of the valid roots
The valid roots are:
- From Case 1: \( x_1 = -1 + 2\sqrt{2} \)
- From Case 2: \( x_4 = 1 - \sqrt{2} \)

Now, we find the sum:
\[ \text{Sum} = x_1 + x_4 = (-1 + 2\sqrt{2}) + (1 - \sqrt{2}) = (-1 + 1) + (2\sqrt{2} - \sqrt{2}) = \sqrt{2} \]

### Final Answer
The sum of the roots of the equation is \( \sqrt{2} \).