What is the average of `(x+y)^(2) and (x-y)^(2)`?

Options

A. `(x+y)/(2)`
B. `xy`
C. `x^(2)-y^(2)`
D. `x^(2)+y^(2)`

Answer

To find the average of \((x+y)^2\) and \((x-y)^2\), we can follow these steps:

### Step 1: Write the formula for the average
The average of two numbers \(a\) and \(b\) is given by the formula:
\[ \text{Average} = \frac{a + b}{2} \]
In our case, \(a = (x+y)^2\) and \(b = (x-y)^2\).

### Step 2: Substitute the values into the formula
Substituting \(a\) and \(b\) into the average formula, we have:
\[ \text{Average} = \frac{(x+y)^2 + (x-y)^2}{2} \]

### Step 3: Expand the squares
Now we need to expand \((x+y)^2\) and \((x-y)^2\):
\[ (x+y)^2 = x^2 + 2xy + y^2 \]
\[ (x-y)^2 = x^2 - 2xy + y^2 \]

### Step 4: Add the expanded forms
Now, we will add the two expanded forms:
\[ (x+y)^2 + (x-y)^2 = (x^2 + 2xy + y^2) + (x^2 - 2xy + y^2) \]
Combining like terms:
\[ = x^2 + 2xy + y^2 + x^2 - 2xy + y^2 \]
\[ = 2x^2 + 2y^2 \]

### Step 5: Substitute back into the average formula
Now we substitute this back into the average formula:
\[ \text{Average} = \frac{2x^2 + 2y^2}{2} \]

### Step 6: Simplify the expression
Now, we can simplify:
\[ \text{Average} = \frac{2(x^2 + y^2)}{2} = x^2 + y^2 \]

### Conclusion
Thus, the average of \((x+y)^2\) and \((x-y)^2\) is:
\[ \boxed{x^2 + y^2} \]