Find the value of 'k' for which `x=3` is a solution of the quadratic equation, `(k+2)x^(2)-kx+6=0`
Hence, find the other root of the equation.

Answer

To solve the problem step by step, we need to find the value of \( k \) such that \( x = 3 \) is a solution of the quadratic equation \( (k + 2)x^2 - kx + 6 = 0 \). Then, we will find the other root of the equation.

### Step 1: Substitute \( x = 3 \) into the equation
Since \( x = 3 \) is a solution, we substitute \( x = 3 \) into the equation:

\[ (k + 2)(3^2) - k(3) + 6 = 0 \]

### Step 2: Simplify the equation
Calculating \( 3^2 \):

\[ (k + 2)(9) - 3k + 6 = 0 \]

Expanding this gives:

\[ 9k + 18 - 3k + 6 = 0 \]

### Step 3: Combine like terms
Combine the terms involving \( k \) and the constant terms:

\[ (9k - 3k) + (18 + 6) = 0 \]

This simplifies to:

\[ 6k + 24 = 0 \]

### Step 4: Solve for \( k \)
Now, we isolate \( k \):

\[ 6k = -24 \]

Dividing both sides by 6:

\[ k = -4 \]

### Step 5: Substitute \( k \) back into the equation
Now that we have \( k = -4 \), we substitute it back into the original quadratic equation:

\[ (-4 + 2)x^2 - (-4)x + 6 = 0 \]

This simplifies to:

\[ -2x^2 + 4x + 6 = 0 \]

### Step 6: Divide the entire equation by -2
To make the equation simpler, divide by -2:

\[ x^2 - 2x - 3 = 0 \]

### Step 7: Factor the quadratic equation
Now, we factor the quadratic:

\[ (x - 3)(x + 1) = 0 \]

### Step 8: Find the roots
Setting each factor to zero gives us the roots:

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Conclusion
Thus, the other root of the equation is:

\[ \text{Other root} = -1 \]