Question

How many solutions exist for the quadratic polynomial in the graph?

(1 point)
real solutions and
non-real solutions

Answer 1

To determine how many solutions exist for a quadratic polynomial based on its graph, we need to look at the position of the graph (specifically a parabola) in relation to the x-axis.

1. **Real Solutions**: Real solutions occur when the parabola intersects the x-axis. Depending on the shape and position of the graph, this can happen in three ways:
- The parabola intersects the x-axis at two distinct points (2 real solutions).
- The parabola touches the x-axis at one point (the vertex is on the x-axis, resulting in 1 real solution).
- The parabola does not intersect the x-axis at all (0 real solutions).

2. **Non-Real Solutions**: Non-real solutions, or complex solutions, occur when the parabola does not intersect the x-axis. In this case, the number of non-real solutions corresponds to the number of pairs of complex conjugate roots:
- If there are no real solutions (the parabola is entirely above or below the x-axis), then there are 2 non-real solutions.

To summarize:
- **2 real solutions** → 0 non-real solutions
- **1 real solution** → 1 pair of non-real solutions
- **0 real solutions** → 2 non-real solutions

If you provide a specific graph or additional context about the quadratic polynomial, I could give a more definitive answer based on that information.