Determine the value of k for which the quadratic equation `4x^(2)-3kx+1=0` has equal roots :

Options

A. `pm((2)/(3))`
B. `pm((4)/(3))`
C. `pm4`
D. `pm6`

Answer

To determine the value of \( k \) for which the quadratic equation \( 4x^2 - 3kx + 1 = 0 \) has equal roots, we can follow these steps:

### Step 1: Understand the condition for equal roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots are equal if the discriminant \( D \) is equal to zero. The discriminant is given by the formula:
\[ D = b^2 - 4ac \]

### Step 2: Identify coefficients
From the given equation \( 4x^2 - 3kx + 1 = 0 \), we can identify the coefficients:
- \( a = 4 \)
- \( b = -3k \)
- \( c = 1 \)

### Step 3: Set up the discriminant equation
Using the discriminant formula, we set up the equation:
\[ D = (-3k)^2 - 4 \cdot 4 \cdot 1 \]
This simplifies to:
\[ D = 9k^2 - 16 \]

### Step 4: Set the discriminant to zero
Since we want the roots to be equal, we set the discriminant to zero:
\[ 9k^2 - 16 = 0 \]

### Step 5: Solve for \( k \)
Now, we solve the equation:
\[ 9k^2 = 16 \]
Dividing both sides by 9 gives:
\[ k^2 = \frac{16}{9} \]
Taking the square root of both sides, we find:
\[ k = \pm \frac{4}{3} \]

### Conclusion
The values of \( k \) for which the quadratic equation has equal roots are:
\[ k = \frac{4}{3} \quad \text{or} \quad k = -\frac{4}{3} \]