If `49x^(2)-b=(7x+(1)/(2))(7x-(1)/(2)),` then the value of b is

Options

A. 0
B. `(1)/(sqrt(2))`
C. `(1)/(4)`
D. `(1)/(2)`

Explanation

To find the value of \( b \) in the equation

\[ 49x^2 - b = (7x + \frac{1}{2})(7x - \frac{1}{2}), \]

we will start by simplifying the right-hand side.

### Step 1: Expand the right-hand side

Using the difference of squares formula, we can expand the right-hand side:

\[ (7x + \frac{1}{2})(7x - \frac{1}{2}) = (7x)^2 - \left(\frac{1}{2}\right)^2. \]

Calculating each term gives:

\[ (7x)^2 = 49x^2, \]
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4}. \]

Thus, we can rewrite the right-hand side:

\[ (7x + \frac{1}{2})(7x - \frac{1}{2}) = 49x^2 - \frac{1}{4}. \]

### Step 2: Set the left-hand side equal to the right-hand side

Now we have:

\[ 49x^2 - b = 49x^2 - \frac{1}{4}. \]

### Step 3: Solve for \( b \)

To find \( b \), we can equate the two sides:

\[ 49x^2 - b = 49x^2 - \frac{1}{4}. \]

Subtract \( 49x^2 \) from both sides:

\[ -b = -\frac{1}{4}. \]

Now, multiply both sides by -1:

\[ b = \frac{1}{4}. \]

### Final Answer

Thus, the value of \( b \) is

\[ \boxed{\frac{1}{4}}. \]

---