If x=4 is one of the roots of the quadratic equation `3x^(2)+kx-2=0`, find the value of k.

Answer

To find the value of \( k \) in the quadratic equation \( 3x^2 + kx - 2 = 0 \) given that \( x = 4 \) is one of its roots, we can follow these steps:

### Step 1: Substitute the root into the equation
Since \( x = 4 \) is a root, we can substitute \( x \) with \( 4 \) in the equation:

\[ 3(4)^2 + k(4) - 2 = 0 \]

### Step 2: Calculate \( 3(4)^2 \)
Calculate \( 3(4)^2 \):

\[ 3(16) = 48 \]

### Step 3: Substitute back into the equation
Now, substitute this value back into the equation:

\[ 48 + 4k - 2 = 0 \]

### Step 4: Simplify the equation
Combine the constants \( 48 - 2 \):

\[ 46 + 4k = 0 \]

### Step 5: Isolate \( k \)
Now, isolate \( k \) by moving \( 46 \) to the other side:

\[ 4k = -46 \]

### Step 6: Solve for \( k \)
Divide both sides by \( 4 \) to find \( k \):

\[ k = \frac{-46}{4} \]

### Step 7: Simplify the fraction
Now simplify \( \frac{-46}{4} \):

\[ k = \frac{-23}{2} \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{-\frac{23}{2}} \]

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