Find the value of k if the quadratic equation `kx(x-2)+6=0` has two equal roots.

Answer

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

### Step 1: Rewrite the Equation
The given equation can be rewritten in standard quadratic form. First, expand the equation:
\[ kx(x-2) + 6 = 0 \]
This simplifies to:
\[ kx^2 - 2kx + 6 = 0 \]
Here, we can identify the coefficients:
- \( a = k \)
- \( b = -2k \)
- \( c = 6 \)

### Step 2: Use the Condition for Equal Roots
For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero gives us:
\[ D = (-2k)^2 - 4(k)(6) = 0 \]

### Step 3: Calculate the Discriminant
Now, calculate the discriminant:
\[ (-2k)^2 = 4k^2 \]
\[ 4(k)(6) = 24k \]
So, we have:
\[ 4k^2 - 24k = 0 \]

### Step 4: Factor the Equation
We can factor out \( 4k \):
\[ 4k(k - 6) = 0 \]

### Step 5: Solve for \( k \)
Setting each factor to zero gives us:
1. \( 4k = 0 \) which implies \( k = 0 \)
2. \( k - 6 = 0 \) which implies \( k = 6 \)

### Conclusion
The values of \( k \) for which the quadratic equation has equal roots are:
\[ k = 0 \quad \text{or} \quad k = 6 \]