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 \( k \)).