Question

Create a function graph to answer the question. Use the function graph to solve for the solutions to the equation x^2+2x=3
.(1 point)
Responses

no solution
no solution

x=2,0
x equals 2 comma 0

x=0,1
x equals 0 comma 1

x=−3,1

Answer 1

To solve the equation \( x^2 + 2x = 3 \) using a function graph, we can first rearrange the equation to set it to zero:

\[
x^2 + 2x - 3 = 0
\]

Next, we can analyze the function defined by:

\[
f(x) = x^2 + 2x - 3
\]

To find the solutions to \( f(x) = 0 \), we would typically plot the graph of this quadratic function.

### Step 1: Identify the components of the function

- The quadratic function is of the form \( f(x) = ax^2 + bx + c \), where \( a = 1, b = 2, c = -3 \).
- The graph is a parabola opening upwards since \( a > 0 \).

### Step 2: Calculate the vertex and the intercepts (optional for sketching)

1. **Vertex**: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} = -\frac{2}{2 \cdot 1} = -1 \).
- Substitute \( x = -1 \) into the function:
\[
f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4
\]
So the vertex is at (-1, -4).

2. **Y-intercept**: Set \( x = 0 \):
\[
f(0) = 0^2 + 2(0) - 3 = -3
\]
The y-intercept is (0, -3).

3. **X-intercepts**: Find the x-intercepts by solving \( f(x) = 0 \):
We can factor \( x^2 + 2x - 3 = (x + 3)(x - 1) = 0 \).
Thus, \( x = -3 \) and \( x = 1 \).

### Step 3: Sketch the Graph

The graph of \( f(x) = x^2 + 2x - 3 \) will have the following key features:
- Vertex at (-1, -4)
- X-intercepts at (-3, 0) and (1, 0)
- Y-intercept at (0, -3)

### Conclusion

From the graph, we can see that the function \( f(x) \) crosses the x-axis at \( x = -3 \) and \( x = 1 \). Therefore, the solutions to the equation \( x^2 + 2x = 3 \) are:

\[
\boxed{x = -3, 1}
\]

Thus the correct response is:

**x = -3, 1**