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# Quadratic equations word problem: box dimensions

Sal solves a volume problem using a quadratic equation. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• I"m a bit confused....? At he talks about how "we don't even have to factor them out, we can just divide them all by 9 to simplify things". But that's actually the ONLY way I would've done it... by dividing it ny 9. I don't really know the other way that he's speaking of.. But it looks like I NEED to know the other way, in case it does need to be "factored out". Can someone pleaseee show me an example of what it would've looked like to factor that equation out? Because he backed out of doing it, I don't know what he WOULD'VE done... I've watched it over and over again. Please help, I'm at a roadblock..
• I would divide the 9 first to get rid of it right of the bat and not have to worry about it.
• At around Sal says 4+0+5=9. How exactly does that work? When can it be applied and when is not wise to apply such a method? Thanks!
• I'm not entirely sure why -9 is not a valid solution. In a cartesian system a length or width of -9 or -5 would be acceptable as it is simply a length of 9 or 5 in the negative direction of the respective axes.
Can someone explain why this is not a valid solution?
i.e.= if we consider the solution as plotting a line on a cartesian plane, the length of the line (or edge of the box) is equal to 9 and is valid in either direction from the origin.
-9---------0---------+9

No?
• I believe that the box presented did not in any specific way relate to cartesian coordinates or graphing. It's like it's length for example spans out in a positive direction both ways, because there is no specific zero point on it. Hope that helps.
• It's cool how Sal can think and solve it so fast. I'm kind of confused. Can someone explain for me in a simpler way?
• The dimensions of a box are length multiplied by width multiplied by height. So the dimensions of the box are (x+4)9x. This equals 405. Distribute the 9x, so it would be (9x) multiplied by (x) plus (9x) multiplied by (4). You would get 9x^2 + 36x=405. Subtract 405 from each side. 9x^2+36x -405=0. Divide each side by 9. x^2+4x-45=0. We now have to factor. 9 and -5 are added together to get 4, which is b. When multiplied they get -45, so it will work. (x+9)(x-5). You can check that by foiling. Now to solve for x we must set both of these to zero. x+9=0, or x=-9. x-5=0 of x=5. The final answer is x= -9,5
• RE: the "Volume of the Box" problem. Shouldn't the length of the box be bigger than its width? In other words, shouldn't the length be x+4 and the width should only equal x. It's maybe a minor point, but I was always told that length is greater than width (l > w). Isn't that true? — MATAPHOBIC
• It doesn't matter what you call the dimensions. It's fairly arbitrary. The math all works out the same no matter what.
• Why can't you have negative distances? I know Sal said we can't have negative distances, but I want to know why.
• Think about it. If you want to go from A to B, and distance equals velocity times time, d=vt and d was negative, that would mean you would get to B before you even left A!

Now, there are a lot of different types of distances, and the math ones, at this level, have one thing in common, they are built on the notion of the "metric space" (all the mathematics you will be doing here at Khan is based on metric space, though talking about it is a more advanced subject). The first rule of metric spaces is d(a,b)≥0, that is, distance is non negative. If it could be negative, then the system we have wouldn't work - you would have conundrums and paradoxes such as arriving before you leave. So negative distance is like dividing by zero, if you try, you just get silly results.

Having said that, a time you might see something that looks like a negative distance is just a way of saying, "to the left." If I had a list of distances that a point had traveled over the number line, such as {2, 5, -4, -2}, I would read that as: the point first moved 2 to the right, then 5 to the right, then 4 to the left, and finally 2 to the left. But as you can see, and as andrew pointed out, even though the point is moving -4, the distance from where it was to where it is is |-4| = 4.

Check out:
https://en.wikipedia.org/wiki/Metric_space
https://en.wikipedia.org/wiki/Distance
• Is there a video that explains the intuition behind this magic?
• It's not magic. It is mathematics! And just think for yourself about this intuition. I just did the same and got it. Read and solve about these topics. I'm sure you would get it. I am telling you to "think" for yourself because it helps a lot and remains in your mind for lifelong!

Good Luck! :)
• how would one factor out 3a^2 -8a -3 . The solution is (3a + 1)(a-3) but I don';t know how to do it. Can someone help me with that one? Thanks
• There is no golden rule when it comes to this, but the fact that it starts with 3a^2 tells you that it's probably going to be in the form of (3a + number)(a - other number). Then you figure out the two numbers that have -3 as a product (luckily there's only two options) and -8a as a sum combined when the other number is multiplied with the 3a in the beginning [-3 * 3a + 1a = -8a]. It takes a while to get a hang of, but it's definitely something that you get better in as you go along.