For what value (s) of a is ` x +(1)/(4) sqrt(x) + a^(2)` a perfect square ?

Options

A. `pm (1)/(18)`
B. `(1)/(8)`
C. `-(1)/(5)`
D. `(1)/(4)`

Answer

To determine the values of \( a \) for which the expression \( x + \frac{1}{4} \sqrt{x} + a^2 \) is a perfect square, we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x + \frac{1}{4} \sqrt{x} + a^2 \]
We can rewrite \( x \) as \( (\sqrt{x})^2 \):
\[ (\sqrt{x})^2 + \frac{1}{4} \sqrt{x} + a^2 \]

### Step 2: Set up the perfect square form
A perfect square trinomial can be expressed in the form:
\[ (\sqrt{x} + b)^2 = (\sqrt{x})^2 + 2b\sqrt{x} + b^2 \]
Here, we want to match our expression with this form. Therefore, we need to identify \( b \) such that:
\[ 2b = \frac{1}{4} \]
and
\[ b^2 = a^2 \]

### Step 3: Solve for \( b \)
From \( 2b = \frac{1}{4} \), we can solve for \( b \):
\[ b = \frac{1}{8} \]

### Step 4: Relate \( b \) to \( a \)
Since \( b^2 = a^2 \), we have:
\[ \left(\frac{1}{8}\right)^2 = a^2 \]
This gives us:
\[ \frac{1}{64} = a^2 \]

### Step 5: Solve for \( a \)
Taking the square root of both sides, we find:
\[ a = \frac{1}{8} \quad \text{or} \quad a = -\frac{1}{8} \]

### Conclusion
Thus, the values of \( a \) for which the expression \( x + \frac{1}{4} \sqrt{x} + a^2 \) is a perfect square are:
\[ a = \frac{1}{8} \quad \text{and} \quad a = -\frac{1}{8} \]

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