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.
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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).
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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.