Find the values of constant c that makes 𝒇(𝒙) continuous: 𝑓(𝑥)={𝑥2−𝑐2 if 𝑥<4𝑐𝑥+20 if 𝑥≥4.

To determine the values of the constant \( c \) that make the function 

\[
f(x) = 
\begin{cases} 
x^2 - c^2 & \text{if } x < 4 \\
4cx + 20 & \text{if } x \geq 4 
\end{cases}
\]

continuous, we need to ensure that the left-hand limit as \( x \) approaches 4 is equal to the right-hand limit at \( x = 4 \), and both of these are equal to \( f(4) \).

1. **Calculate the left-hand limit as \( x \) approaches 4:**

\[
\lim_{x \to 4^-} f(x) = 4^2 - c^2 = 16 - c^2
\]

2. **Calculate the right-hand limit as \( x \) approaches 4:**

\[
\lim_{x \to 4^+} f(x) = 4c(4) + 20 = 16c + 20
\]

3. **Set the limits equal to each other for continuity:**

\[
16 - c^2 = 16c + 20
\]

4. **Rearrange the equation:**

\[
-c^2 - 16c + 16 - 20 = 0 \\
-c^2 - 16c - 4 = 0
\]

Multiplying the entire equation by -1 gives:

\[
c^2 + 16c + 4 = 0
\]

5. **Use the quadratic formula to solve for \( c \):**

\[
c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-16 \pm \sqrt{16^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{-16 \pm \sqrt{256 - 16}}{2} = \frac{-16 \pm \sqrt{240}}{2}
\]

Simplifying further:

\[
c = \frac{-16 \pm 4\sqrt{15}}{2} = -8 \pm 2\sqrt{15}
\]

Thus, the values of \( c \) that make \( f(x) \) continuous are:

\[
c = -8 + 2\sqrt{15} \quad \text{and} \quad c = -8 - 2\sqrt{15}
\]

### Summary of Results:
- \( c = -8 + 2\sqrt{15} \)
- \( c = -8 - 2\sqrt{15} \)

Would you like any further details or have any questions? Here are some related questions you might consider:

1. What is the value of \( c \) approximately?
2. How does changing the value of \( c \) affect the function's graph?
3. Can you show how to plot the function \( f(x) \) for different values of \( c \)?
4. What happens to the continuity of \( f(x) \) if \( c \) is not one of these values?
5. Can you explain how the quadratic formula works in this context?

**Tip:** Always verify your solutions by substituting them back into the original conditions to check for continuity.

Explore how to determine the values of c that ensure the continuity of the piecewise function f(x) defined as x^2 - c^2 for x < 4 and 4cx + 20 for x ≥ 4. This solution involves limits and the quadratic formula, suitable for advanced high school students.

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