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 −6 negative 6 0 0 1 1 4

Answer 1

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

Starting with the original equation:

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

Let's move all terms to one side of the equation:

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

This simplifies to:

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

To make it easier to solve, we'll multiply the entire equation by -1:

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

Now, we can factor the quadratic equation:

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

Setting each factor to zero gives us the solutions:

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

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

So the solutions to the equation \( -x^2 + 5x = -x + 5 \) are \( x = 1 \) and \( x = 5 \).

Now, we need to find which value of \( k \) in the table corresponds to these solutions. According to the responses, the possible \( k \) values are: \( -6, 0, 1, 4, 5 \).

The values from the equation \( -x^2 + 5x = -x + 5 \) that are in the list are \( 1 \) and \( 5 \).

Thus, the value of \( k \) that will be a solution to the equation is:

**1** and **5**.