# Complex roots Part 1

Carlos has not had the best morning. First, he was out of milk for his cereal. Then he splattered mustard his shirt while frantically making his lunch for the day. By the time he left his house, he realized he might even miss the bus.

As Carlos rounded the last corner, he saw his bus. He instantly started running after it at full speed, 10m/s. At the same time, the bus revved up its engines and started accelerating at a rate of 5m/s2. The bus already had a 10 meter head start on Carlos. Will Carlos catch the bus?

Problems like these are typical in a pre-calculus class. The usual solution is to model the distance between the bus and Carlos as a function of time. Then use some method to solve the equation. Rather than delving too deep into the mathematical process, I want to highlight a few situations graphically with you.

Suppose Carlos can run really fast. It is possible that he could catch up to the bus and pass the bus. However, eventually the bus would get up to speed and would pass Carlos. This situation can be modelled by the following graph. The red line is the position of the bus and the blue line is the position of Carlos.

Maybe Carlos isn’t the best runner. Maybe he just barely catches up with the bus. The graph below shows this possibility.

Finally, maybe the bus is too fast, or had too much of a head start, or Carlos had a stomach cramp. Whatever the case, Carlos misses his bus. This situation is graphed below.

The first and second possibilities are usually the ones that are emphasized in class. You can use a formula (the quadratic formula) to find out what happens to Carlos, how long he has to run to catch the bus, where exactly he catches it, and so on. The third situation is simply left as “no solution.” But that is not entirely true. Although Carlos does not catch up with the bus, he does get closer to it. In fact, there is a point when he gets really close, but then the bus zooms away. If you work through the formula, you will see that that you actually find out how close Carlos got to the bus and at what time. The only problem is that we have to cheat a bit. Stay tuned for part 2.