Question

The graph shows the equation y=x+2
in red and the equation y=x2
in blue.







Which of the following shows the solution(s) for this system of equations?

(1 point)
Responses

(4,2)
open paren 4 comma 2 close paren

There are no solutions
There are no solutions

(0,0)
and (0,2)
open paren 0 comma 0 close paren and open paren 0 comma 2 close paren

(−1,1)
and (2,4)

Answer 1

To find the solution(s) for the system of equations \(y = x + 2\) (red) and \(y = x^2\) (blue), we need to determine the points where the two graphs intersect.

1. Set the equations equal to each other:
\[
x + 2 = x^2
\]

2. Rearrange the equation to form a standard quadratic equation:
\[
x^2 - x - 2 = 0
\]

3. Factor the quadratic equation:
\[
(x - 2)(x + 1) = 0
\]

4. Solving for \(x\) gives:
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]

5. Now, substitute these \(x\) values into either original equation to find the corresponding \(y\) values.

For \(x = 2\):
\[
y = 2 + 2 = 4
\]
So, one solution is \((2, 4)\).

For \(x = -1\):
\[
y = -1 + 2 = 1
\]
So, the other solution is \((-1, 1)\).

Therefore, the solutions are:
- \( (2, 4) \)
- \( (-1, 1) \)

The correct response from the options given is:
**(−1,1) and (2,4)**.