Square root of `3/2 (x - 1) + sqrt(2x^2 - 7 x - 4)` is

Options

A. `1/2 (sqrt(2x + 1) + sqrt(x - 4))`
B. `1/(sqrt(2))(sqrt(2x + 1) + sqrt(x - 4))`
C. `1/(sqrt2) (sqrt(2x - 1) + sqrt(x + 4))`
D. `1/2 (sqrt(2x - 1) + sqrt(x + 4))`

Answer

To solve the expression \( \sqrt{\frac{3}{2}(x - 1) + \sqrt{2x^2 - 7x - 4}} \), we will follow these steps:

### Step 1: Simplify the Expression Inside the Square Root
We start with the expression:
\[ \sqrt{\frac{3}{2}(x - 1) + \sqrt{2x^2 - 7x - 4}} \]

### Step 2: Analyze the Quadratic Expression
Next, we need to simplify the quadratic expression \( \sqrt{2x^2 - 7x - 4} \). We can try to factor or complete the square for the quadratic:
\[ 2x^2 - 7x - 4 \]

### Step 3: Complete the Square for the Quadratic
To complete the square, we can rewrite \( 2x^2 - 7x - 4 \):
1. Factor out the 2 from the first two terms:
\[ 2(x^2 - \frac{7}{2}x) - 4 \]
2. Complete the square inside the parentheses:
\[ x^2 - \frac{7}{2}x = \left(x - \frac{7}{4}\right)^2 - \left(\frac{7}{4}\right)^2 \]
3. Substitute back:
\[ 2\left(\left(x - \frac{7}{4}\right)^2 - \frac{49}{16}\right) - 4 \]
4. Simplify:
\[ 2\left(x - \frac{7}{4}\right)^2 - \frac{49}{8} - 4 = 2\left(x - \frac{7}{4}\right)^2 - \frac{49 + 32}{8} = 2\left(x - \frac{7}{4}\right)^2 - \frac{81}{8} \]

### Step 4: Substitute Back into the Original Expression
Now we substitute this back into our original expression:
\[ \sqrt{\frac{3}{2}(x - 1) + \sqrt{2\left(x - \frac{7}{4}\right)^2 - \frac{81}{8}}} \]

### Step 5: Combine Terms
This expression can be complex, but we can combine the terms under the square root. We will need to evaluate the expression further based on the values of \( x \).

### Step 6: Final Expression
After simplification, we can express the result in a more manageable form. The final expression will depend on the simplifications made in previous steps.

### Final Answer
The final answer can be expressed as:
\[ \sqrt{\frac{3x - 3}{2} + \sqrt{2\left(x - \frac{7}{4}\right)^2 - \frac{81}{8}}} \]