If `(x + 7957 xx 7965)` is a perfect sqaure, then find the value of x ?

Options

A. 1
B. 16
C. 9
D. 25

Answer

To solve the problem, we need to determine the value of \( x \) such that \( x + 79,057 \times 79,065 \) is a perfect square.

### Step 1: Rewrite the terms
We can express \( 79,057 \) and \( 79,065 \) in terms of a common number:
- \( 79,057 = 79,061 - 4 \)
- \( 79,065 = 79,061 + 4 \)

### Step 2: Use the difference of squares
Now, we can use the identity for the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]
In our case, we have:
\[ 79,057 \times 79,065 = (79,061 - 4)(79,061 + 4) = 79,061^2 - 4^2 \]
This simplifies to:
\[ 79,061^2 - 16 \]

### Step 3: Substitute back into the equation
Now we substitute back into our original expression:
\[ x + 79,057 \times 79,065 = x + (79,061^2 - 16) \]

### Step 4: Set the expression to be a perfect square
For \( x + (79,061^2 - 16) \) to be a perfect square, we can set:
\[ x + 79,061^2 - 16 = k^2 \]
for some integer \( k \).

### Step 5: Rearrange the equation
Rearranging gives us:
\[ x = k^2 - 79,061^2 + 16 \]

### Step 6: Find a suitable \( k \)
To ensure that \( x \) is a perfect square, we can choose \( k = 79,061 \):
\[ x = 79,061^2 - 79,061^2 + 16 = 16 \]

### Conclusion
Thus, the value of \( x \) that makes the expression a perfect square is:
\[ \boxed{16} \]