Question

x −x^2 +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 −x^2 +5x=−x+5 ?(1 point) Responses 1 1 0 0 −6 negative 6 4 4

Answer 1

To find the value of \( k \) that satisfies the equation \(-x^2 + 5x = -x + 5\), we first need to simplify this equation. First, rearranging the equation gives:

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

This simplifies to:

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

Now, multiplying everything by \(-1\) (to make the leading coefficient positive) results in:

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

Next, we can factor this quadratic equation:

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

Setting each factor equal to zero gives us the potential solutions:

\[
x - 1 = 0 \implies x = 1
\]
\[
x - 5 = 0 \implies x = 5
\]

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

Now, we need to check the table of values provided:

The table indicates the following values for \( x \):

- 0
- 0
- 1 (which is one of our solutions)
- 4
- 2
- 6
- 3
- 6
- 4
- 4
- 5 (which is the other solution)
- 0

Any of these values that match \( k \) which should be either 1 or 5 will be a solution to the given equation.

Thus, the correct responses for \( k \) that are solutions are 1 and 5. If you have to choose from the available options, then:

The response is **1** or **5** (depending on which one they were asking for in terms of \( k \)).