If `4pxy = (x + 2y) ^(2) - (x - 2y) ^(2),` then what will be the value of p ?

Options

A. `0.5`
B. `0.25`
C. 4
D. 2

Answer

To solve the equation \( 4pxy = (x + 2y)^2 - (x - 2y)^2 \) and find the value of \( p \), we can follow these steps:

### Step 1: Expand the squares on the right-hand side
We will use the formula for the square of a binomial, which states that \( (a + b)^2 = a^2 + 2ab + b^2 \) and \( (a - b)^2 = a^2 - 2ab + b^2 \).

- For \( (x + 2y)^2 \):
\[ (x + 2y)^2 = x^2 + 2 \cdot x \cdot 2y + (2y)^2 = x^2 + 4xy + 4y^2 \]

- For \( (x - 2y)^2 \):
\[ (x - 2y)^2 = x^2 - 2 \cdot x \cdot 2y + (2y)^2 = x^2 - 4xy + 4y^2 \]

### Step 2: Substitute the expanded forms into the equation
Now we substitute these expansions back into the original equation:
\[ 4pxy = (x^2 + 4xy + 4y^2) - (x^2 - 4xy + 4y^2) \]

### Step 3: Simplify the right-hand side
Now, we simplify the right-hand side:
\[ = x^2 + 4xy + 4y^2 - x^2 + 4xy - 4y^2 \]
The \( x^2 \) and \( 4y^2 \) terms cancel out:
\[ = 4xy + 4xy = 8xy \]

### Step 4: Set the two sides equal to each other
Now we have:
\[ 4pxy = 8xy \]

### Step 5: Divide both sides by \( 4xy \) (assuming \( xy \neq 0 \))
\[ p = \frac{8xy}{4xy} = 2 \]

### Conclusion
Thus, the value of \( p \) is \( 2 \).

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