Question
The Usefulness of Graphs Quick Check
5 of 55 of 5 Items
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
4
4
1
1
0
0
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Answers
GPT-4o mini
To find the value of \( k \) in the table that satisfies the equation \( -x^2 + 5x = -x + 5 \), we first need to rearrange the equation:
\[
-x^2 + 5x + x - 5 = 0
\]
This simplifies to:
\[
-x^2 + 6x - 5 = 0
\]
Multiplying through by -1 gives us:
\[
x^2 - 6x + 5 = 0
\]
Next, we can factor this quadratic equation:
\[
(x - 1)(x - 5) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 1 = 0 \implies x = 1
\]
\[
x - 5 = 0 \implies x = 5
\]
Now, we check the values of \( k \) given in the options. The quadratic equations have solutions of \( x = 1 \) and \( x = 5 \).
Checking the provided options for \( k \):
- -6
- 4
- **1**
- 0
Since one of the solutions we found is \( x = 1 \), the value of \( k \) that is a solution to the equation \( -x^2 + 5x = -x + 5 \) is:
\[
\boxed{1}
\]
\[
-x^2 + 5x + x - 5 = 0
\]
This simplifies to:
\[
-x^2 + 6x - 5 = 0
\]
Multiplying through by -1 gives us:
\[
x^2 - 6x + 5 = 0
\]
Next, we can factor this quadratic equation:
\[
(x - 1)(x - 5) = 0
\]
Setting each factor equal to zero gives us the solutions:
\[
x - 1 = 0 \implies x = 1
\]
\[
x - 5 = 0 \implies x = 5
\]
Now, we check the values of \( k \) given in the options. The quadratic equations have solutions of \( x = 1 \) and \( x = 5 \).
Checking the provided options for \( k \):
- -6
- 4
- **1**
- 0
Since one of the solutions we found is \( x = 1 \), the value of \( k \) that is a solution to the equation \( -x^2 + 5x = -x + 5 \) is:
\[
\boxed{1}
\]