To find the real solutions of the equation x² + 2x + 2 = 0 by graphing, we first need to understand what this equation represents. The equation is a quadratic function, which can be represented by the parabola that opens upwards.
1. **Understand the Function**: The standard form of a quadratic equation is ax² + bx + c = 0. Here, our coefficients are a = 1, b = 2, and c = 2.
2. **Graph the Parabola**: To graph the quadratic, we can find the vertex and plot a few points. The vertex can be found using the formula x = -b/(2a). In our case, that gives us:
- Vertex x-coordinate: x = -2/(2*1) = -1
- To find the y-coordinate, substitute x = -1 back into the equation:
- y = (-1)² + 2(-1) + 2 = 1 – 2 + 2 = 1
3. **Plotting Points**: Now we can plot the vertex at (-1, 1) and a few other points around it to get a clearer view of the parabola:
- For x = 0: y = 2
- For x = -2: y = 2
- For x = -3: y = 5
4. **Drawing the Graph**: Once we plot these points, we can draw the curve of the parabola. The graph will show that the parabola does not intersect the x-axis at any point.
5. **Conclusion**: Since the parabola does not touch the x-axis, there are no real solutions to the equation x² + 2x + 2 = 0. Instead, the solutions are complex.