Finding Real Solutions by Graphing
To find the real solutions of the quadratic equation x² + 2x + 2 = 0 through graphing, we first need to understand the basic characteristics of the parabola represented by this equation.
Step 1: Understand the Equation
The given equation can be rewritten in standard form, showing the values of a, b, and c: where a = 1, b = 2, and c = 2. This means the parabola opens upwards, since a is positive.
Step 2: Determine the Vertex
The vertex of the parabola can be found using the formula x = -b/(2a). Plugging in our values:
- x = -2/(2 * 1) = -1
To find the y-coordinate of the vertex, substitute x = -1 back into the equation:
- y = (-1)² + 2(-1) + 2 = 1 – 2 + 2 = 1
Thus, the vertex is at the point (-1, 1).
Step 3: Graph the Equation
Plot the vertex on the Cartesian plane. Since the vertex is at (-1, 1) and the parabola opens upwards, it will be symmetric around the line x = -1. Draw the parabola by choosing additional points on either side of the vertex. We can calculate a few points:
- For x = -2:
y = (-2)² + 2(-2) + 2 = 4 – 4 + 2 = 2 → Point (-2, 2) - For x = 0:
y = (0)² + 2(0) + 2 = 2 → Point (0, 2)
Plotting these points along with the vertex gives a clearer shape of the parabola.
Step 4: Analyze the Graph
From the graph, we can observe that the parabola does not cross the x-axis. Since the y-coordinate of the vertex is 1, which is above the x-axis, it signifies that there are no real roots for this equation.
Conclusion
The quadratic equation x² + 2x + 2 = 0 does not have real solutions, as confirmed by the graph which shows that the parabola does not intersect the x-axis.