`(723+1992)^2-(723-1992)^2 ÷723 × 1992=?`

Options

A. 4
B. 33
C. 6
D. 1

Answer

To solve the expression `(723 + 1992)^2 - (723 - 1992)^2 ÷ 723 × 1992`, we will follow these steps:

1. **Identify Variables**: Let \( a = 723 \) and \( b = 1992 \). This simplifies our expression.

2. **Rewrite the Expression**: The expression can be rewritten as:
\[ (a + b)^2 - (a - b)^2 \div (a \times b) \]

3. **Apply the Difference of Squares Formula**: We can use the identity \( x^2 - y^2 = (x - y)(x + y) \) where \( x = (a + b) \) and \( y = (a - b) \):
\[ (a + b)^2 - (a - b)^2 = [(a + b) - (a - b)][(a + b) + (a - b)] \]
Simplifying each part:
- \( (a + b) - (a - b) = 2b \)
- \( (a + b) + (a - b) = 2a \)

Therefore, we have:
\[ (a + b)^2 - (a - b)^2 = 2b \cdot 2a = 4ab \]

4. **Substitute Back into the Expression**: Now substitute \( 4ab \) back into the expression:
\[ 4ab \div (ab) \]

5. **Simplify the Expression**: The \( ab \) in the numerator and denominator cancels out:
\[ 4ab \div ab = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]