If the equation `9x^(2)+6kx+4=0` has equal roots then k=?

Options

A. 2 or 0
B. `-2" or "0`
C. 2 or -2
D. 0 only

Explanation

To find the value of \( k \) for which the quadratic equation \( 9x^2 + 6kx + 4 = 0 \) has equal roots, we can follow these steps:

### Step 1: Identify the standard form of a quadratic equation
A quadratic equation is generally given in the form:
\[ ax^2 + bx + c = 0 \]
In our case, we have:
- \( a = 9 \)
- \( b = 6k \)
- \( c = 4 \)

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant (\( \Delta \)) must be equal to zero. The discriminant is given by:
\[ \Delta = b^2 - 4ac \]
Setting the discriminant to zero gives us:
\[ b^2 - 4ac = 0 \]

### Step 3: Substitute the values of \( a \), \( b \), and \( c \)
Substituting the values we identified:
\[ (6k)^2 - 4 \cdot 9 \cdot 4 = 0 \]

### Step 4: Simplify the equation
Calculating \( (6k)^2 \):
\[ 36k^2 - 144 = 0 \]

### Step 5: Rearrange the equation
Now, we can rearrange the equation to isolate \( k^2 \):
\[ 36k^2 = 144 \]

### Step 6: Divide both sides by 36
\[ k^2 = \frac{144}{36} \]
\[ k^2 = 4 \]

### Step 7: Take the square root of both sides
Taking the square root gives us:
\[ k = \pm 2 \]

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

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