Question

Create a function graph to answer the question. Use the function graph to solve for the solutions to the equation x2+2x=3
.(1 point)
Responses

no solution
no solution

x=0,1
x equals 0 comma 1

x=2,0
x equals 2 comma 0

x=−3,1

Answer 1

To solve the equation \(x^2 + 2x = 3\), we can first rewrite it in standard quadratic form:

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

Next, we'll create a function from the left side of the equation:

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

We'll find the solutions to the equation by determining where the graph of the function \(f(x)\) intersects the x-axis (where \(f(x) = 0\)).

To sketch the graph of \(f(x)\), we can find the vertex and the y-intercept.

1. **Finding the Vertex**:
The vertex of a parabola \(ax^2 + bx + c\) is given by the x-coordinate \(x = -\frac{b}{2a}\):
- Here, \(a = 1\), \(b = 2\) so:
\[
x = -\frac{2}{2 \cdot 1} = -1
\]
To find the y-coordinate of the vertex, substitute \(x = -1\) into the function:
\[
f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4
\]
The vertex is at the point \((-1, -4)\).

2. **Finding the Y-Intercept**:
The y-intercept occurs when \(x = 0\):
\[
f(0) = 0^2 + 2(0) - 3 = -3
\]
The y-intercept point is \((0, -3)\).

3. **Finding the X-Intercepts**:
We can find the x-intercepts by factoring the quadratic equation \(x^2 + 2x - 3\):
\[
(x + 3)(x - 1) = 0
\]
Setting each factor to zero gives us:
\[
x + 3 = 0 \Rightarrow x = -3
\]
\[
x - 1 = 0 \Rightarrow x = 1
\]

The x-intercepts are at the points \((-3, 0)\) and \((1, 0)\).

### Summary of the Solutions
The solutions to the original equation \(x^2 + 2x = 3\) are where \(f(x) = 0\):
- \(x = -3\)
- \(x = 1\)

Thus, the correct response is:
**x = −3, 1**

If providing this in a multiple-choice format:

- **x = −3, 1** (correct response)
- x = 0, 1
- x = 2, 0
- no solution

### Conclusion
The solutions to the equation \(x^2 + 2x = 3\) are \(x = -3\) and \(x = 1\).