Use the tables to answer the question.

Question

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

4
4

1
1

0
0

−6

Answer 1

To find the value of $k$ such that $-x^2+5x=-x+5$, we will set both expressions equal to each other and solve for $x$.

Starting with the equation:

$$-x^2+5x=-x+5$$

Let's reorganize the equation:

$$-x^2+5x+x-5=0$$

This simplifies to:

$$-x^2+6x-5=0$$

To make it easier to work with, we can multiply the entire equation by -1:

$$x^2-6x+5=0$$

Now we will factor the quadratic equation:

$$(x-5)(x-1)=0$$

Setting each factor equal to zero gives us the solutions:

$$x-5=0\Rightarrow x=5$$ $$x-1=0\Rightarrow x=1$$

Thus, the solutions to the equation are $x=1$ and $x=5$.

Now, we need to check which value of $k$ corresponds to one of the solutions from the second table. In that table:

• For $k=4$, the corresponding output (value) is $3$.
• For $k=3$, the corresponding output (value) is $2$.
• For $k=5$, the corresponding output (value) is $0$.
• For $k=7$, the corresponding output (value) is $-2$.
• The column for $-2$ yields $0$ for $k=5$, which is one of our solutions.

Thus, both $k=1$ and $k=5$ are solutions to the original equation.

The available responses are:

• 4
• 1
• 0
• -6

Therefore, the answer is 1, since it is one of the values representing a solution to the equation $-x^2+5x=-x+5$.