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 solve the equation \( 49x^2 - b = (7x + \frac{1}{2})(7x - \frac{1}{2}) \), we will follow these steps:

### Step 1: Recognize the Identity
We can use the identity for the difference of squares:
\[ (a + b)(a - b) = a^2 - b^2 \]
In our case, let \( a = 7x \) and \( b = \frac{1}{2} \).

### Step 2: Apply the Identity
Using the identity, we can rewrite the right-hand side:
\[ (7x + \frac{1}{2})(7x - \frac{1}{2}) = (7x)^2 - \left(\frac{1}{2}\right)^2 \]
Calculating this gives:
\[ (7x)^2 - \left(\frac{1}{2}\right)^2 = 49x^2 - \frac{1}{4} \]

### Step 3: Substitute Back into the Equation
Now we substitute this back into the original equation:
\[ 49x^2 - b = 49x^2 - \frac{1}{4} \]

### Step 4: Simplify the Equation
Next, we can simplify the equation:
\[ 49x^2 - b = 49x^2 - \frac{1}{4} \]
To isolate \( b \), we can subtract \( 49x^2 \) from both sides:
\[ -b = -\frac{1}{4} \]

### Step 5: Solve for \( b \)
Now, we multiply both sides by -1 to solve for \( b \):
\[ b = \frac{1}{4} \]

### Final Answer
Thus, the value of \( b \) is:
\[ \boxed{\frac{1}{4}} \]