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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.

Video transcript

The volume of a box is 405 cube units, or I guess cubic units. So they just want to keep it general. It could've been in cubic feet, or cubic meters, or cubic centimeters, or cubic miles. Who knows? They just want to keep it as units, keep it as general as possible. The length is x units, the width is x plus 4 units, and the height is 9 units. So let me draw this box here. Let me draw a little box here, so we have a nice little visualization. So they tell us, that the length is x. Maybe we could call this the length right there. They say the width is x plus 4, and the height is 9 of this box. In units, what are the dimensions of the box? Well, they also tell us that the volume is 405. So the volume, 405-- let me do it this way. So if we wanted to calculate the volume, what would it be? Well it would be the width-- it would be x plus 4 times the length -- times x-- times 9. That's, literally, the volume of the box. Now they also tell us that the volume of the box is 405 cubic units, is equal to 405. So now we just solve for x. So what do we get here? If we distribute this x into this x plus 4. Actually, if we distribute a 9x. Let me just rewrite it. This is the same thing as 9x times x plus 4 is equal to 405. 9x times x is equal to 9x squared. 9x times 4 is equal to 36x, is equal to 405. Now we want our quadratic expression to be equal to 0. So let's subtract 405 from both sides of this equation. So when you do that, your right-hand side equals 0, and your left-hand side is 9x squared plus 36x minus 405. Now, is there any common factor to these numbers right here? Well 405, 4 plus 0 plus 5 is 9, so that is divisible by 9. So all of these are divisible by 9. Let's just figure out what 405 divided by 9 is. So 9 goes into 405-- 9 goes into 40 4 times. 4 times 9 is 36. Subtract you get 45. 9 goes into 45 5 times. 5 times 9 is 45. Subtract, you get 0. So it goes 45 times. So if we factor out a 9 here, we get 9 times x squared-- actually even better, you don't even have to factor out of 9. If you think about it, you can divide both sides of this equation by 9. So if you can divide all of the terms by 9, it won't change the equation. You're doing the same thing to both sides of equations, which we've learned long ago is a very valid thing to do. So here you get x squared-- if you just had this expression, here, and someone told you to factor it, then you'd have to factor out the 9. But because this is an equation, it equals 0, let's just divide everything by 9. It'll simplify things. So you get x squared plus 4x minus 45 is equal to 0. And now we can try to factor this right here. And this fits the pattern, where we don't have a leading 1 out here. So we don't even have to do it by grouping. You just have to think, what 2 numbers, when I take their product I get negative 45, and when I take their sum, I get positive 4. They are 4 apart. 1 has to be positive, 1 has to be negative. Their positive versions have to be 4 apart. Because when you take the sum, you are really taking their difference because 1 of them is negative. So let's think about it. When you have positive 9 and negative 5, I think that'll work. Right? Positive 9 plus negative 5 is 4. And when you take the product, you get negative 45. So you have x plus 9 times x minus 5 is equal to 0. Just factored it out. And we've seen this before. If you have 2 numbers, when you take their product that equals 0, that means 1 of these numbers at least has to be equal 0. So this means that x plus 9 is equal to 0. Scroll down a little bit. x plus 9 is equal 0, or x minus 5 is equal 0. So if we subtract 9 from this equation right there, you get x is equal to negative 9, or if you add 5 to both sides of this equation, here. You get x is equal to 5. So these are both possible values of x right here. So the box, if you take x is equal to negative 9, well, x equal to negative 9 won't work. Because if you but negative 9 here, you're going to have a box that has a width of negative 5, a length of negative 9, and a height of 9. And if we're talking about our reality, we don't have negative distances like this. That can't be the length or the width. So x equals negative 9 isn't appropriate for this problem Because in this problem we need to have positive dimensions. So let's see what happens with x equals 5. If x equals 5, x plus 4 is 9, and this dimension right here is going to be 5. And that seems pretty reasonable for our reality. And let's verify that this does end up with a volume of 405. 9 times 5 is 45 times 9 is indeed 405. We just figured that out over here, that 45 times 9 is 405. So we're done.