The area of a square is given by `A = x^(2) + 4x + 4`, then the diagonal of the square is

Options

A. (x-2)
B. (x+2)
C. `sqrt(2)(x-sqrt(2))`
D. `sqrt(2) (x+2)`

Answer

To find the diagonal of a square given its area, we can follow these steps:

### Step 1: Identify the area of the square
The area of the square is given by the expression:
\[ A = x^2 + 4x + 4 \]

### Step 2: Factor the area expression
We need to factor the quadratic expression \( x^2 + 4x + 4 \). We can rewrite it as:
\[ A = (x + 2)^2 \]
This is because \( x^2 + 4x + 4 \) can be factored as \( (x + 2)(x + 2) \).

### Step 3: Determine the side length of the square
Since the area of the square is equal to the side length squared, we can equate:
\[ \text{Side} = x + 2 \]

### Step 4: Use the Pythagorean theorem to find the diagonal
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = \sqrt{(\text{side})^2 + (\text{side})^2} \]
Substituting the side length:
\[ d = \sqrt{(x + 2)^2 + (x + 2)^2} \]
This simplifies to:
\[ d = \sqrt{2(x + 2)^2} \]

### Step 5: Simplify the expression for the diagonal
We can take the square root out:
\[ d = \sqrt{2} \cdot (x + 2) \]

### Conclusion
Thus, the diagonal of the square is:
\[ d = (x + 2) \sqrt{2} \]