Determine the value of m for which ` 4 x + ( sqrt(x))/( 6) + (m^(2))/( 4) ` is a perfect square

Options

A. `(1)/(24)`
B. `(1)/(12)`
C. 12
D. 24

Answer

To determine the value of \( m \) for which the expression \( 4x + \frac{\sqrt{x}}{6} + \frac{m^2}{4} \) is a perfect square, we can follow these steps:

### Step 1: Substitute \( x \) with \( y^2 \)
We start by substituting \( x \) with \( y^2 \). This gives us:
\[ 4y^2 + \frac{y}{6} + \frac{m^2}{4} \]

### Step 2: Identify coefficients
In the expression \( 4y^2 + \frac{y}{6} + \frac{m^2}{4} \), we can identify:
- \( A = 4 \)
- \( B = \frac{1}{6} \)
- \( C = \frac{m^2}{4} \)

### Step 3: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( Ay^2 + By + C \) is given by:
\[ D = B^2 - 4AC \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = \left(\frac{1}{6}\right)^2 - 4 \cdot 4 \cdot \frac{m^2}{4} \]
This simplifies to:
\[ D = \frac{1}{36} - 4m^2 \]

### Step 4: Set the discriminant to zero
For the expression to be a perfect square, the discriminant must equal zero:
\[ \frac{1}{36} - 4m^2 = 0 \]

### Step 5: Solve for \( m^2 \)
Rearranging the equation gives:
\[ 4m^2 = \frac{1}{36} \]
Dividing both sides by 4:
\[ m^2 = \frac{1}{144} \]

### Step 6: Find \( m \)
Taking the square root of both sides, we find:
\[ m = \pm \frac{1}{12} \]

Thus, the values of \( m \) for which the expression is a perfect square are:
\[ m = \frac{1}{12} \quad \text{or} \quad m = -\frac{1}{12} \]

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