Interpret quadratic models: Factored form | Algebra I | Khan Academy

We're told that Rodrigo watches a helicopter take off from a platform. The height of the helicopter in meters above the ground, t minutes after takeoff, is modeled by... and we see this function right over here. Rodrigo wants to know when the helicopter will land on the ground, so pause this video and see if you can figure that out.

All right, now let's think about this together. So, let's just imagine actually what the graph of this function looks like. It'll also help us imagine what's going on with the helicopter.

So, our horizontal axis, this is t, time in minutes, and then our vertical axis is height. So, height as a function of time, and maybe I just write it like this: I'll just write height, and this is given in meters above the ground.

Now, I don't know exactly what the graph looks like, but given that I have a negative coefficient on my quadratic term, I know that it is a downward-opening parabola like that. It says that the helicopter takes off of a platform, so however high the platform is, then it takes off, and it's going to do something like this.

I don't know exactly what the graph looks like, but probably something like this. Now, if they asked us what is the highest point of the helicopter and at what time does it happen, then we'd want to figure out what the vertex is of this parabola.

But that's not what they're asking; they're asking when does a helicopter land on the ground. That's this time right over here. So, if we wanted to find the vertex, we would want to put this into vertex form, but here we want to figure out when does that function equal zero. We want to find a zero of this quadratic right over here.

So, the best way that I can think about doing it is to try to factor it, try to set this thing equal to zero, and then factor it and then see what t values make that equal to zero.

So let me do that. I say negative three t squared plus 24 t plus 60, remember we care when our height is equal to zero, equals zero. So let's see, maybe the first thing I would do, just to simplify this second-degree term a little bit, let's just divide both sides by negative three.

If we did that, this would become t squared. 24 divided by negative three is negative eight, negative eight t. Sixty divided by negative three is negative twenty, and then zero divided by negative three is of course still zero.

Now, can I think of two numbers whose product is negative 20? So, they would have to have different signs in order to get a negative product, and whose sum is negative 8.

So, let's see, what about negative 10 and 2? That seems to work. So, I could write this as t minus 10 times t plus 2 is equal to 0.

And so, in order to make this entire expression equal to 0, either one of these could be equal to 0. So, either t minus 10 is equal to 0, or t plus 2 is equal to 0.

And of course, on the left here, I can add 10 to both sides, so either t equals 10, or I could subtract 2 from both sides here, t is equal to negative 2.

So, there's two places where the function is equal to zero: one at time t equals negative two and one at time t is equal to ten. Now, we're assuming we're dealing with positive time here; we don't know what the helicopter was doing before the takeoff, so we wouldn't really think about this.

So, what we really care about is that t is equal to 10 minutes. That's when the helicopter is right over there, and actually, we know at t equals zero these two terms become zero. We know it takes off at 60 meters, it goes up; if we figured out the vertex, we would know how high it went.

But then it starts going back down, and in 10 minutes after takeoff, it is back at zero, back on the ground.