If ` (3x+(1)/(2))(3x-(1)/(2))=9x^2-p` then the value of `p` is

Options

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

Explanation

To solve the equation \((3x + \frac{1}{2})(3x - \frac{1}{2}) = 9x^2 - p\), we will follow these steps:

### Step 1: Recognize the expression
We see that the left-hand side of the equation is in the form of \((a + b)(a - b)\), where \(a = 3x\) and \(b = \frac{1}{2}\).

**Hint:** Identify the structure of the expression. It resembles the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\).

### Step 2: Apply the difference of squares formula
Using the difference of squares formula, we can rewrite the left-hand side:
\[ (3x + \frac{1}{2})(3x - \frac{1}{2}) = (3x)^2 - \left(\frac{1}{2}\right)^2 \]

**Hint:** Remember that \((a + b)(a - b) = a^2 - b^2\).

### Step 3: Calculate \(a^2\) and \(b^2\)
Now we calculate:
\[ (3x)^2 = 9x^2 \]
and
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

**Hint:** Square both \(3x\) and \(\frac{1}{2}\) to find \(a^2\) and \(b^2\).

### Step 4: Substitute back into the equation
Now substituting these values back into the equation gives us:
\[ 9x^2 - \frac{1}{4} = 9x^2 - p \]

**Hint:** Substitute the calculated squares into the expression.

### Step 5: Equate and solve for \(p\)
Since both sides of the equation have \(9x^2\), we can equate the remaining parts:
\[ -\frac{1}{4} = -p \]
This implies:
\[ p = \frac{1}{4} \]

**Hint:** Isolate \(p\) by moving terms across the equation.

### Final Answer
Thus, the value of \(p\) is \(\frac{1}{4}\).

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