The Usefulness of Graphs Quick Check

Question

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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Answer 1

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:

\[ \boxed{1} \]