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Solving equations graphically (2 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

In the last video, we estimated the solution to e to the x is equal to 1 over x times x minus 1 times x minus 2 using a calculator. We got a first rough estimate by just looking at this graph, and then we tried values out to truly zero in on, or get close to the x value where this is true. What I now want to do is actually just use the graphing functionality of this calculator to try to estimate the solution graphically. So let's go to Graph, and what I'm going to try to do is graph both of these functions. So the first function-- let me clear this-- the first one is e of x, which on the graphing calculator will be y1. And that's going to be e to the x power, and then the second one, y2, will be r of x, which is going to be 1 divided by x times x minus 1 times x minus 2. And so, let's see. I have to close this parentheses as well. So I've entered it in the graph. And I care about where we zoom in, so I really want to zoom into this part of the graph right over here. So let me go to the Range. So, actually there's also a Zoom functionality that I could use, but let's-- actually, let me do that. That could be fun, so let's just graph it. Actually, let's just see what range it's graphed at right now. Let's see. What we would care about, let's start with a rough approximation, just to see that this is indeed the same graph. So let's start with x going from 0 up to-- I don't know-- 3, so this would be this part of the graph right over here. And then the x scale is 1. That's what they'll mark off, every 1. We could even mark off every 0.5 if we want. Like this one is marked off every 0.5. And the y minimum, let's go from 0 to-- on this range, actually it goes pretty high, and the way this is graphed, it goes all the way up to, looks like 10, so I'll go 10. I'll leave the y scale as 1. They mark it off every 1 right over here. And now let's graph this thing. And that was e of x, and now it's graphing r of x. And you see it indeed looks pretty similar to what we have here. Now what we care about is this point, or on our calculator, this point right over here. We want to figure out what x value-- what is the x-coordinate of this point of intersection? This is when our two functions are equal to each other. So let me zoom in on this. I think I can use this Box functionality. So it essentially lets me construct a box around this, and it's going to zoom into that box. So I'm going to get as tight in on this as I can go. So if I press, I can get even tighter on it. So if I press Enter, now I can define the other corner of the box, so that's pretty good. I'm going to zoom in, press Enter, and that's zoomed in on to that little teeny box. So that was e of x, and now it's going to graph r of x. So now let me try to trace the graph. So let's see. Trace, so it's letting me trace e of x. And let's see. If I look at my x values decrease-- so at this point, e of x is still higher than r of x. And if we get right over here-- so 2.056, we see that r of x is above e of x. We just see that graphically, and then we're left of the point of intersection. And then we're still left of the point of intersection. Now we're right of the point of intersection, so it looks like the point of intersection is between 2.057 and 2.059. And so in the previous video when we said where our estimate was 2.06, we were definitely within 0.01 of the point of intersection. If we did want to get even more precise, we could zoom in more, and I encourage you if you've got a graphing calculator like this to actually try that out.