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To solve the problem of finding the value of \( k \) that makes the function \( f(x) \) continuous, we need to ensure that the left-hand limit and right-hand limit of the function are equal at the point where the piecewise definition changes, which is \( x = 9 \).

The function is defined as:

\[
f(x) = 
\begin{cases} 
k^2 - 2x, & x < 9 \\
kx, & x \geq 9 
\end{cases}
\]

For the function to be continuous at \( x = 9 \):

1. The left-hand limit as \( x \) approaches 9 must equal the right-hand limit.
2. The value of the function at \( x = 9 \) must equal these limits.

### Step-by-Step Solution

1. **Left-hand limit as \( x \to 9^- \):**
   \[
   \lim_{{x \to 9^-}} f(x) = k^2 - 2(9) = k^2 - 18
   \]

2. **Right-hand limit as \( x \to 9^+ \):**
   \[
   \lim_{{x \to 9^+}} f(x) = k(9) = 9k
   \]

3. **Setting the limits equal for continuity:**
   \[
   k^2 - 18 = 9k
   \]

4. **Solving the quadratic equation:**

   Rewrite the equation:

   \[
   k^2 - 9k - 18 = 0
   \]

   This is a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -9 \), and \( c = -18 \).

   Use the quadratic formula:

   \[
   k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   Substituting the values of \( a \), \( b \), and \( c \):

   \[
   k = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(-18)}}{2(1)}
   \]

   \[
   k = \frac{9 \pm \sqrt{81 + 72}}{2}
   \]

   \[
   k = \frac{9 \pm \sqrt{153}}{2}
   \]

5. **Calculating the possible values of \( k \):**

   The two possible values of \( k \) are:

   \[
   k_1 = \frac{9 + \sqrt{153}}{2}, \quad k_2 = \frac{9 - \sqrt{153}}{2}
   \]

These are the values of \( k \) that make the function continuous.

### Conclusion

The function \( f(x) \) is continuous for the values of \( k \):

\[
k = \frac{9 + \sqrt{153}}{2} \quad \text{or} \quad k = \frac{9 - \sqrt{153}}{2}
\]

Would you like more details or have any questions? 

### Related Questions:

1. How do you determine the continuity of a function at a specific point?
2. What is the significance of continuity in a real-world scenario?
3. How does a discontinuity affect the graph of a function?
4. Can you provide an example of a function that is continuous everywhere except at one point?
5. What are the differences between continuous and differentiable functions?

### Tip: 

Always remember to check both left-hand and right-hand limits when determining continuity at a point.

Find the values of k that make the function continuous: f(x) = {k^2 - 2x, for x < 9; kx, for x >= 9}

Solve for the values of k that make the piecewise function continuous at x = 9. This solution involves concepts of continuity and solving quadratic equations. Suitable for high school students in Grades 10-12.

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