If “from the square of a half the sum of two numbers we subtract the square of a half their difference ", the result is the :

Options

A. sum of the two numbers
B. quotient of the two numbers
C. difference of the two numbers
D. product of the two numbers

Answer

To solve the problem step by step, let's break down the statement and perform the necessary algebraic manipulations.

### Step 1: Define the Numbers
Let the two numbers be \( x \) and \( y \).

### Step 2: Find the Sum and Difference
- The sum of the two numbers is \( x + y \).
- The difference of the two numbers is \( x - y \).

### Step 3: Calculate Half of the Sum and Half of the Difference
- Half of the sum is \( \frac{x + y}{2} \).
- Half of the difference is \( \frac{x - y}{2} \).

### Step 4: Square the Half of the Sum
Now, we need to square half of the sum:
\[ \left(\frac{x + y}{2}\right)^2 = \frac{(x + y)^2}{4} = \frac{x^2 + 2xy + y^2}{4} \]

### Step 5: Square the Half of the Difference
Next, we square half of the difference:
\[ \left(\frac{x - y}{2}\right)^2 = \frac{(x - y)^2}{4} = \frac{x^2 - 2xy + y^2}{4} \]

### Step 6: Subtract the Square of the Half of the Difference from the Square of the Half of the Sum
Now, we subtract the square of the half of the difference from the square of the half of the sum:
\[ \frac{x^2 + 2xy + y^2}{4} - \frac{x^2 - 2xy + y^2}{4} \]

### Step 7: Combine the Fractions
To combine the fractions, we have:
\[ \frac{(x^2 + 2xy + y^2) - (x^2 - 2xy + y^2)}{4} \]

### Step 8: Simplify the Expression
Now, simplify the numerator:
\[ (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = x^2 + 2xy + y^2 - x^2 + 2xy - y^2 = 4xy \]
Thus, we have:
\[ \frac{4xy}{4} = xy \]

### Conclusion
The result of the expression is:
\[ xy \]
This means the result is the product of the two numbers \( x \) and \( y \).

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