Find the value of 'a' for which `x^2+sqrt(3)x +(a^(2))/(4)` is a perfect square.

Options

A. `sqrt(3)`
B. `2sqrt(3)`
C. `3sqrt(3)`
D. `4sqrt(3)`

Answer

To find the value of 'a' for which the expression \( x^2 + \sqrt{3}x + \frac{a^2}{4} \) is a perfect square, we can follow these steps:

### Step 1: Identify the coefficients
The given expression is in the form of a quadratic polynomial \( Ax^2 + Bx + C \), where:
- \( A = 1 \) (coefficient of \( x^2 \))
- \( B = \sqrt{3} \) (coefficient of \( x \))
- \( C = \frac{a^2}{4} \) (constant term)

### Step 2: Use the condition for a perfect square
For a quadratic polynomial to be a perfect square, the discriminant must be equal to zero. The discriminant \( D \) is given by the formula:
\[ D = B^2 - 4AC \]
Setting \( D = 0 \) gives us the condition for the expression to be a perfect square.

### Step 3: Substitute the values into the discriminant formula
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (\sqrt{3})^2 - 4 \cdot 1 \cdot \frac{a^2}{4} \]
This simplifies to:
\[ D = 3 - a^2 \]

### Step 4: Set the discriminant to zero
To find the value of \( a \), we set the discriminant equal to zero:
\[ 3 - a^2 = 0 \]

### Step 5: Solve for \( a^2 \)
Rearranging the equation gives:
\[ a^2 = 3 \]

### Step 6: Find the value of \( a \)
Taking the square root of both sides results in:
\[ a = \sqrt{3} \quad \text{or} \quad a = -\sqrt{3} \]

Thus, the values of \( a \) for which the expression is a perfect square are \( a = \sqrt{3} \) and \( a = -\sqrt{3} \).

### Final Answer
The possible values of \( a \) are \( \sqrt{3} \) and \( -\sqrt{3} \).

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