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# Quadratic equations word problem: box dimensions

CCSS Math: HSA.CED.A.1, HSA.REI.B.4, HSA.REI.B.4b, HSA.SSE.B.3, HSA.SSE.B.3a

## 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.