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

Answer

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

### Step 1: Rewrite the equation
Start with the given equation:
\[ kx(x-2) + 6 = 0 \]
Expanding this, we get:
\[ kx^2 - 2kx + 6 = 0 \]

### Step 2: Identify coefficients
In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), we can identify:
- \( a = k \)
- \( b = -2k \)
- \( c = 6 \)

### Step 3: Use the condition for equal roots
For a quadratic equation to have two equal roots, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero for our equation:
\[ (-2k)^2 - 4(k)(6) = 0 \]

### Step 4: Simplify the discriminant equation
Calculating the discriminant:
\[ 4k^2 - 24k = 0 \]

### Step 5: Factor the equation
We can factor out \( 4k \):
\[ 4k(k - 6) = 0 \]

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

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

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