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

Answer

To find the value of \( k \) when \( x = 3 \) is one root of the quadratic equation \( x^2 - 2kx - 6 = 0 \), follow these steps:

1. **Substitute \( x = 3 \) into the quadratic equation:**

Given equation: \( x^2 - 2kx - 6 = 0 \)

Substitute \( x = 3 \):
\[ (3)^2 - 2k(3) - 6 = 0 \]

2. **Simplify the equation:**

Calculate \( 3^2 \):
\[ 9 - 2k(3) - 6 = 0 \]

3. **Combine like terms:**

Simplify \( -2k(3) \) to \( -6k \):
\[ 9 - 6k - 6 = 0 \]

Combine \( 9 \) and \( -6 \):
\[ 3 - 6k = 0 \]

4. **Solve for \( k \):**

Isolate \( k \) by moving \( 3 \) to the other side of the equation:
\[ -6k = -3 \]

Divide both sides by \( -6 \):
\[ k = \frac{-3}{-6} \]

Simplify the fraction:
\[ k = \frac{1}{2} \]

So, the value of \( k \) is \( \frac{1}{2} \).