If one root of the quadratic equation `6x^(2)-x-k=0" is" (2)/(3)`, then find the value of k.

Answer

To find the value of \( k \) in the quadratic equation \( 6x^2 - x - k = 0 \) given that one root is \( \frac{2}{3} \), we can follow these steps:

### Step 1: Substitute the root into the equation
Since \( \frac{2}{3} \) is a root of the equation, we can substitute \( x = \frac{2}{3} \) into the equation:

\[ 6\left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right) - k = 0 \]

### Step 2: Calculate \( \left(\frac{2}{3}\right)^2 \)
First, we calculate \( \left(\frac{2}{3}\right)^2 \):

\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]

### Step 3: Substitute back into the equation
Now substitute \( \frac{4}{9} \) back into the equation:

\[ 6 \cdot \frac{4}{9} - \frac{2}{3} - k = 0 \]

### Step 4: Simplify \( 6 \cdot \frac{4}{9} \)
Calculate \( 6 \cdot \frac{4}{9} \):

\[ 6 \cdot \frac{4}{9} = \frac{24}{9} = \frac{8}{3} \]

### Step 5: Substitute and simplify the equation
Now we have:

\[ \frac{8}{3} - \frac{2}{3} - k = 0 \]

### Step 6: Combine the fractions
Combine \( \frac{8}{3} - \frac{2}{3} \):

\[ \frac{8}{3} - \frac{2}{3} = \frac{6}{3} = 2 \]

### Step 7: Set the equation to solve for \( k \)
Now we have:

\[ 2 - k = 0 \]

### Step 8: Solve for \( k \)
Rearranging gives:

\[ k = 2 \]

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

\[ \boxed{2} \]