In a nonlinear system, two curved graphs can meet more than twice. The cue is a question asking how many distinct real pairs satisfy both equations, rather than asking for their coordinates. Graph both equations in Desmos and count every intersection. Looking at only one half of the graph can make you miss solutions on the other half.
Hints
- Hint 1
An intersection is a point shared by two graphs, so its coordinates satisfy both equations. Since the question asks for a count, what should you count on the graph?
- Hint 2
A parabola is a U-shaped curve. The other equation graphs a circle. Look for places where the two curves cross, both above and below the -axis.
- Hint 3
The parabola is symmetric across the -axis: opposite -values give the same . If you spot a crossing on one side, check for its mirror on the other.
Step-by-step
Count intersections in Desmos
Step 1Graph the parabola
Type into Desmos. It draws a parabola, a U-shaped curve. Click its lowest point to see . The curve rises on both sides of the -axis, so keep both sides in view when you add the circle.
- Step 2
Count the points shared with the circle
Type on the next line. Desmos draws the entire circle in this window. Click the shared points: approximately , , , and . Each intersection is one pair that satisfies both equations. There are four distinct real solutions . Choice A.