If `(x + 7954 xx 7956)` be a square number, then the value of x is

Options

A. 1
B. 16
C. 9
D. 4

Answer

To solve the problem, we need to find the value of \( x \) such that \( x + 7954 \times 7956 \) is a perfect square.

### Step-by-step Solution:

1. **Understand the Expression**:
We need to analyze the expression \( x + 7954 \times 7956 \). First, we calculate \( 7954 \times 7956 \).

2. **Calculate \( 7954 \times 7956 \)**:
We can use the difference of squares:
\[ 7954 \times 7956 = (7955 - 1)(7955 + 1) = 7955^2 - 1^2 = 7955^2 - 1 \]

3. **Substitute Back into the Expression**:
Now, substitute this back into the expression:
\[ x + 7954 \times 7956 = x + (7955^2 - 1) = x + 7955^2 - 1 \]

4. **Set the Expression Equal to a Perfect Square**:
We want this expression to be a perfect square, so we set:
\[ x + 7955^2 - 1 = y^2 \]
Rearranging gives:
\[ x = y^2 - 7955^2 + 1 \]

5. **Use the Difference of Squares**:
The expression \( y^2 - 7955^2 \) can be factored:
\[ y^2 - 7955^2 = (y - 7955)(y + 7955) \]
Therefore:
\[ x = (y - 7955)(y + 7955) + 1 \]

6. **Finding Suitable Values for \( y \)**:
For \( x \) to be a non-negative integer, \( y \) must be chosen such that \( (y - 7955)(y + 7955) \) is a non-negative integer.

7. **Choose \( y = 7955 + 1 \)**:
Let’s try \( y = 7956 \):
\[ x = (7956 - 7955)(7956 + 7955) + 1 = (1)(15911) + 1 = 15911 + 1 = 15912 \]

8. **Final Value of \( x \)**:
Thus, the value of \( x \) that makes the expression a perfect square is:
\[ x = 1 \]

### Conclusion:
The value of \( x \) is \( \boxed{1} \).