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To complete the table, we need to use the given functions:

- \( f(x) = -2x + 6 \)
- \( g(x) = 2(x - 1)^2 \)

We'll substitute the values of \( x \) into both functions and fill in the missing parts of the table.

Let's go through each \( x \)-value step by step:

1. **For \( x = 0 \):**
   - \( f(0) = -2(0) + 6 = 6 \) (already provided)
   - \( g(0) = 2(0 - 1)^2 = 2(1)^2 = 2 \) (already provided)

2. **For \( x = 1 \):**
   - \( f(1) = -2(1) + 6 = -2 + 6 = 4 \)
   - \( g(1) = 2(1 - 1)^2 = 2(0)^2 = 0 \)

3. **For \( x = 2 \):**
   - \( f(2) = -2(2) + 6 = -4 + 6 = 2 \)
   - \( g(2) = 2(2 - 1)^2 = 2(1)^2 = 2 \)

4. **For \( x = 3 \):**
   - \( f(3) = -2(3) + 6 = -6 + 6 = 0 \)
   - \( g(3) = 2(3 - 1)^2 = 2(2)^2 = 8 \)

5. **For \( x = 4 \):**
   - \( f(4) = -2(4) + 6 = -8 + 6 = -2 \) (already provided)
   - \( g(4) = 2(4 - 1)^2 = 2(3)^2 = 18 \) (already provided)

Now, the completed table is:

\[
\begin{array}{|c|c|c|}
\hline
x & f(x) & g(x) \\
\hline
0 & 6 & 2 \\
1 & 4 & 0 \\
2 & 2 & 2 \\
3 & 0 & 8 \\
4 & -2 & 18 \\
\hline
\end{array}
\]

### Solution to \( f(x) = g(x) \)

To find where \( f(x) = g(x) \), look for where the values in the table match:

- At \( x = 2 \), \( f(2) = g(2) = 2 \).

Thus, the equation \( f(x) = g(x) \) has a solution at \( x = 2 \).

Would you like any further details or explanations?

Here are 5 related questions to expand on this:

1. What does it mean when \( f(x) = g(x) \) geometrically?
2. How would you solve \( f(x) = g(x) \) algebraically, without a table?
3. What are the properties of the functions \( f(x) \) and \( g(x) \)?
4. How would the graph of \( f(x) \) and \( g(x) \) look?
5. Can you find other points where \( f(x) \) and \( g(x) \) intersect, if they do?

**Tip:** Always plot the functions to better visualize where they intersect.

Complete the table of values for the functions f(x) = -2x + 6 and g(x) = 2(x - 1)^2. Based on the values in the table, where does the equation f(x) = g(x) have a solution?

Complete the table of values for f(x) = -2x + 6 and g(x) = 2(x - 1)^2, then find where f(x) = g(x). This problem covers concepts of linear and quadratic functions, and solving for equal values. Suitable for students in Grades 8-10.

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