Find the square root of:
`49x^(4) -(7x^(2))/2 + (673 x^(2))/16 - (3x)/2 + 9`

Options

A. `7x^(2) - (3x)/7 + 3`
B. `7x^(2) - (7x)/3 +3`
C. `7x^(2) - (2x)/3 + 3`
D. `7x^(2) - x/4 + 3`

Answer

To find the square root of the expression \( 49x^4 - \frac{7x^2}{2} + \frac{673x^2}{16} - \frac{3x}{2} + 9 \), we will follow these steps:

### Step 1: Rewrite the Expression
First, we rewrite the expression clearly:
\[ 49x^4 - \frac{7x^2}{2} + \frac{673x^2}{16} - \frac{3x}{2} + 9 \]

### Step 2: Combine Like Terms
Next, we will combine the terms involving \( x^2 \):
- The terms \( -\frac{7x^2}{2} \) and \( \frac{673x^2}{16} \) need a common denominator to combine. The common denominator is 16.
\[ -\frac{7x^2}{2} = -\frac{56x^2}{16} \]
Thus, we can rewrite the expression as:
\[ 49x^4 + \left(-\frac{56x^2}{16} + \frac{673x^2}{16}\right) - \frac{3x}{2} + 9 \]
This simplifies to:
\[ 49x^4 + \frac{617x^2}{16} - \frac{3x}{2} + 9 \]

### Step 3: Identify the Perfect Square
We can try to express the polynomial in a form that resembles a perfect square. We look for a structure of the form \( (a - b + c)^2 \).

Let:
- \( a = 7x^2 \)
- \( b = \frac{x}{4} \)
- \( c = 3 \)

### Step 4: Expand the Perfect Square
Using the identity \( (a - b + c)^2 = a^2 - 2ab + b^2 + 2ac - 2bc + c^2 \):
- \( a^2 = (7x^2)^2 = 49x^4 \)
- \( b^2 = \left(\frac{x}{4}\right)^2 = \frac{x^2}{16} \)
- \( c^2 = 3^2 = 9 \)
- \( -2ab = -2(7x^2)(\frac{x}{4}) = -\frac{14x^3}{4} = -\frac{7x^3}{2} \)
- \( 2ac = 2(7x^2)(3) = 42x^2 \)
- \( -2bc = -2(\frac{x}{4})(3) = -\frac{3x}{2} \)

### Step 5: Combine the Expanded Terms
Now, we combine these terms:
\[ 49x^4 + \left(-\frac{7x^3}{2} + 42x^2 + 9\right) \]
This matches our original expression.

### Step 6: Write the Square Root
Thus, we can express the original polynomial as:
\[ (7x^2 - \frac{x}{4} + 3)^2 \]
Taking the square root gives us:
\[ \sqrt{49x^4 - \frac{7x^2}{2} + \frac{673x^2}{16} - \frac{3x}{2} + 9} = 7x^2 - \frac{x}{4} + 3 \]

### Final Answer
The square root of the expression is:
\[ 7x^2 - \frac{x}{4} + 3 \]