The area of a rhombus is `(1)/(2)x^(2) + 2x + (3)/(2)`. Then, its smaller diagonal is

Options

A. x+2
B. x+3
C. x+1
D. None of these

Answer

To find the smaller diagonal of the rhombus given its area, we can follow these steps:

### Step 1: Write down the area of the rhombus
The area of the rhombus is given as:
\[ \text{Area} = \frac{1}{2}x^2 + 2x + \frac{3}{2} \]

### Step 2: Use the formula for the area of a rhombus
The area of a rhombus can also be expressed in terms of its diagonals \(d_1\) and \(d_2\):
\[ \text{Area} = \frac{1}{2} d_1 d_2 \]
Setting the two expressions for the area equal gives:
\[ \frac{1}{2} d_1 d_2 = \frac{1}{2}x^2 + 2x + \frac{3}{2} \]

### Step 3: Eliminate the fraction
To eliminate the fraction, multiply both sides by 2:
\[ d_1 d_2 = x^2 + 4x + 3 \]

### Step 4: Factor the quadratic expression
Now, we need to factor the quadratic expression \(x^2 + 4x + 3\). We look for two numbers that multiply to 3 (the constant term) and add to 4 (the coefficient of \(x\)):
\[ x^2 + 4x + 3 = (x + 1)(x + 3) \]

### Step 5: Identify the diagonals
From the factored form, we can identify the diagonals:
\[ d_1 = x + 1 \quad \text{and} \quad d_2 = x + 3 \]

### Step 6: Determine the smaller diagonal
To find the smaller diagonal, we compare \(d_1\) and \(d_2\):
\[ d_1 = x + 1 \quad \text{and} \quad d_2 = x + 3 \]
Since \(x + 1 < x + 3\), the smaller diagonal is:
\[ \text{Smaller diagonal} = d_1 = x + 1 \]

### Final Answer
The smaller diagonal of the rhombus is:
\[ \boxed{x + 1} \]