If the equation `2x^2 + kx +3= 0` has two equal roots, then the value of k is

Options

A. `pm 2sqrt3`
B. `pm sqrt6`
C. `pm 2sqrt6`
D. none of these

Answer

To find the value of \( k \) for which the equation \( 2x^2 + kx + 3 = 0 \) has two equal roots, we can use the condition that the discriminant of a quadratic equation must be zero for the roots to be equal.

### Step 1: Identify the coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 2 \)
- \( b = k \)
- \( c = 3 \)

### Step 2: Write the discriminant condition
The discriminant \( D \) of a quadratic equation is given by:
\[ D = b^2 - 4ac \]
For the roots to be equal, we set the discriminant to zero:
\[ D = 0 \]

### Step 3: Substitute the coefficients into the discriminant
Substituting the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = k^2 - 4 \cdot 2 \cdot 3 \]
This simplifies to:
\[ D = k^2 - 24 \]

### Step 4: Set the discriminant to zero
Now, we set the discriminant equal to zero:
\[ k^2 - 24 = 0 \]

### Step 5: Solve for \( k \)
Rearranging gives:
\[ k^2 = 24 \]
Taking the square root of both sides results in:
\[ k = \pm \sqrt{24} \]
This can be further simplified:
\[ k = \pm 2\sqrt{6} \]

### Final Answer
Thus, the values of \( k \) for which the equation has two equal roots are:
\[ k = 2\sqrt{6} \quad \text{or} \quad k = -2\sqrt{6} \]