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:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
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 \( -x^2 + 5x = -x + 5 \).
1 answer