Question

Use the tables to answer the question.

x −x2+5x
0 0
1 4
2 6
3 6
4 4
5 0
x −x+5
−1
6
k 4
3 2
5 0
7 −2
What value of k in the table will be a solution to the equation −x2+5x=−x+5
?(1 point)
Responses

1
1

4
4

0
0

−6
negative 6
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Answer 1

To find the value of \( k \) that is a solution to the equation \( -x^2 + 5x = -x + 5 \), we can first rearrange the equation:

\[
-x^2 + 5x + x - 5 = 0
\]

This simplifies to:

\[
-x^2 + 6x - 5 = 0
\]

Next, we can multiply through by \(-1\) to make it easier to solve:

\[
x^2 - 6x + 5 = 0
\]

Now we can factor the quadratic:

\[
(x - 1)(x - 5) = 0
\]

Setting each factor equal to zero gives us the solutions:

\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
\[
x - 5 = 0 \quad \Rightarrow \quad x = 5
\]

Next, we look for these values (1 and 5) in the second table where \( k \) is located:

\[
\begin{array}{c|c}
x & -x + 5 \\
\hline
-1 & 6 \\
k & 4 \\
3 & 2 \\
5 & 0 \\
7 & -2 \\
\end{array}
\]

From the second table, we see that \( k \) corresponds to the output for \( x \). Since our solutions to the equation are \( x = 1 \) and \( x = 5 \), we notice that:

- The value \( 5 \) is given (resulting in \( 0 \)).
- The value \( k \) which results in \( 4 \) must correspond to \( x = 1 \).

Thus, the solution to the equation \( -x^2 + 5x = -x + 5 \) implies:

The correct value of \( k \) is:

\[
\boxed{4}
\]