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Solving equations graphically (1 of 2)

Sal solves the equation e^x=1/[x(x-1)(x-2)] by considering the graphs of y=e^x and y=1/[x(x-1)(x-2)]. Created by Sal Khan.

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Video transcript

Graphs of e of x equals e to the x and r of x is equal to 1 over x times x minus 1 times x minus 2 are shown below. Estimate the solution to e to the x-- so that's e of x-- being equal to essentially r of x within 0.01. So we want to figure out for what value does e of x equal r of x? And they want us to estimate it. We can either just try to get as close as we can from this graph. They want us to be within 0.01. And we can also use a calculator, kind of try numbers out to hopefully zero in on this point right over here, where e of x is equal to r of x. So what I want to do is, let me draw a little table here. Let's try out some x values. And then for each of these x values, let's see where we land on e of x and where we land on r of x, and then we can decide whether we are too high or too low. And I encourage you to pause this video before I actually go ahead and do this, and try to do this on your own. But I do suggest using some form of calculator or, well, probably a calculator. I'm assuming you've given a go at it, and now I will attempt it. Now just eyeballing it-- and eyeballing it is helpful, because that'll give us kind of our first order approximation of at what x value are these two functions equal. If I just look at this graph the way it's drawn, it looks like this is pretty close to 2.1. It looks like when x is 2.1, both of those functions look pretty close to-- I don't know. This looks like about 7.7 or 7.8 or something like that. But let's figure out what they're doing. So let's see, when x is equal to 2.1-- get my calculator out-- when x is equal to 2.1, well, e of x is just e to the x power. So e to the 2.1 power is equal to 8.166. Let me write that down, 8.166. And what is r of x? r of x is 1 divided by x, so that's going to be 2.1. Times x minus 1. Well, that's going to be 1.1, so that's times 1.1. Times x minus 2. Well, that's just going to be 0.1, times 0.1. And that is equal to 4-- did I do that right? No, that can't be. 2.1 over-- 2.1 times 1.1 times 0.1. 1 over all of that. 4.32? Let's see, 2.1 r of x is 4 point-- I guess that's possible. Actually, that looks right, because r of x declines so sharply right over here. So it's actually, at 2.1 where actually r of x is actually closer to right over here, give or take. So it's equal to 4.32. So 2.1 e of x is actually a much larger value than r of x. So e of x is clearly too high. r of x is already dropped a good bit by then. If I were to go all the way down to 2, at 2 it looks like actually r of x kind of spikes up. It just goes to infinity as we approach 2. So we're not going to go all the way down to 2, but why don't we lower this a little bit. Why don't we try 2.05? So 2.05, what is e of x? e of x is e to the x, right? So e to the 2.05 power gets us 7.76-- I'll round it, 8-- 7.768. Approximately 7-- actually, all these are approximate, so I'll just write 7.768. And what is r of x? I'll just keep rounding to the thousands. Here, well, we didn't have to round too much just because that was so far off, but I'll put it there. Actually, it was 329, so I could-- let me write it this way-- 3290. So let me throw that 9 here, just so everything-- we evaluate the function to thousandths. So let's evaluate r of x, when we're at 2.05, it's going to be 1 divided by x, which is now 2.05, times x minus 1, which is 1.05, times x minus 2, which is 0.05. And that gets us to 9.29-- I'll round to 2-- 9.292, so 9.292. So now we're on this side, where r of x is roughly right over here and it's more than e of x, which is at 7.7, which is right around here. So now our x value is too low. So maybe let's see if we can go a little bit higher. And let's try to go roughly halfway between these two, but I don't want to get too precise, because you have to get to the nearest hundredth. So let's go to 2.07. So e to the 2.07 is equal to 7.925 if I round it, 7.925. I want to do all this in green just to be color consistent. And now let's evaluate r of x at that same value. So 1 divided by x, which is 2.07, times that minus 1, which is 1.07, times that minus 2, which is 0.07, which gives us 6.44, I guess we could say 6.450. So at 2.05 that was too low, 2.07 is too high. Now, r of x has dropped below e of x. So we know the right answer is in between these two numbers, and so if we select 2.06 that's definitely going to be within 0.01 of the right answer. So I would go with 2.06 is definitely going to be within the 0.01 of the correct solution to this. But just for fun, let's actually just try it out. So e to the 2.06 is 7.84-- I guess we could round to 6. And if we were to evaluate r of x, it's 1 divided by 2.06 times that minus 1, which is 1.06, times 0.06. It gets us to 7.632. So we're also getting pretty close, but our precision that they gave, they don't say that they have to be within each other of that, they say, let's estimate the solution. So there's some actual precise solution to this right over here, some x value, where these are actually equal to each other. That's the x value, which gives us this point of intersection. We just have to get within 0.01 of that x value, and 2.06 definitely does the trick.