Main content

### Course: Algebra (all content) > Unit 9

Lesson 5: Solving quadratics by factoring- Solving quadratics by factoring
- Solving quadratics by factoring
- Quadratics by factoring (intro)
- Solving quadratics by factoring: leading coefficient ≠ 1
- Quadratics by factoring
- Solving quadratics using structure
- Solve equations using structure
- Quadratic equations word problem: triangle dimensions
- Quadratic equations word problem: box dimensions
- Solving quadratic equations by factoring (old)
- Solving quadratics by factoring review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# 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....? At2:47he 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..(11 votes)
- I would divide the 9 first to get rid of it right of the bat and not have to worry about it.(6 votes)

- At around2:30Sal 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!(5 votes)
- He is using a divisibility test to quickly determine if something is divisible by 9.

Numbers divisible by 9 have digits that sum up to 9. For instance,

63 => 6+3=9

108 => 1+0+8=9

666 => 6+6+6=18 => 1+8=9

99882 => 9+9+8+8+2=36 => 3+6=9

Check out this video and possible others: https://www.khanacademy.org/math/arithmetic/factors-multiples/divisibility_tests/v/divisibility-tests-for-2--3--4--5--6--9--10(18 votes)

- 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?(5 votes)- 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.(11 votes)

- 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?(6 votes)
- 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(4 votes)

- 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(3 votes)
- It doesn't matter what you call the dimensions. It's fairly arbitrary. The math all works out the same no matter what.(7 votes)

- Why can't you have negative distances? I know Sal said we can't have negative distances, but I want to know why.(4 votes)
- 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(4 votes)

- Is there a video that explains the intuition behind this magic?(4 votes)
- 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! :)(5 votes)

- 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(3 votes)
- 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.(4 votes)

- Is there any way to find a single value for a variable in a quadratic equation (x)? I understand why there would be two possibilities for the answers (except in "reality" questions like volume or area) but I was wondering if all quadratic equations have two solutions and therefore no single answer.(3 votes)
- That's a good question. In general, there are always two answers, but in some cases the two answers might be the same, so there is essentially only one answer. This would happen in a case like x^2 - 4x + 4 = 0; factoring this we get (x-2)(x-2)=0 or (x-2)^2=0, which has two solutions of x=2. This happens when the quadratic factors into two of the same factor. Since a quadratic is the equation for a parabola, and the solutions to the quadratic equations are the points where the parabola crosses the x axis, this single solution corresponds to a parabola that is just "sitting" on the x axis, touching it at only one point. Sorry, that's hard to explain in text, but I hope at least the first part helps.(4 votes)

- What is the way to solve trinomial quadratic equations if the only common factor of the terms is 1 and there is a coefficient other than 1?(2 votes)
- You can complete the square, factor by grouping, or use the quadratic equation.(2 votes)

## 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.