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Current time:0:00Total duration:11:54

We're on problem 32. What are the solutions to the
equation 1 plus 1 over x squared is equal to 3 over x? So at first. This looks like
a pretty daunting equation. You have these x's in the
denominator and x squared in the denominator. But I think we can simplify it
if we can just get rid of these x squares in
the denominator. The easiest way to do
that is to multiply everything by x squared. So let's multiply both sides of
this equation by x squared. Then we'll get x squared times
1 is x squared, x squared times 1 over x squared,
that's just 1. Then x squared times 3 over x,
that's 3x squared over x. x squared divided by x is just
x, so that is equal to 3x. We can subtract 3x from both
sides and you get x squared minus 3x plus 1 is equal to 0. This is a simple quadratic. It's not obvious that
you can factor it. In fact, two numbers when you
multiply them equal 1, and then when you add them
equal minus 3. I'm guessing it might
even be imaginary. It will probably not be
imaginary, but it's just a strange number. So let's use the quadratic
equation. When in doubt, use the
quadratic equation. So minus B, this is B right? B is this 3 right there,
negative 3. B is negative 3. So minus B is going to be plus
3, plus or minus the square root of B squared. Minus 3 squared is 9, minus 4
times A, which is 1, times C which is 1. So it's minus 4, all
of that over 2A. A is 1 so it's just over 2. That is equal to 3/2 plus
or minus the square root of 5 over 2. I just separated these out
because I'm looking at the choices and it seems
like they did that. So you could've said that's 3/2
plus square root of 5 over 2 or 3/2 minus the square
root of 5 over 2. I just did that because
it seems like that's how they write it. That is choice A. Next problem, 33. I think this one actually might
be good to copy and paste the problem. Let me see if I can do this. OK, there are two numbers with
the following properties. Let me write down the properties
and let me copy and paste it for you. OK, I've copied it. Let me go here. Then I've pasted it for you. All right. So there's two numbers with
the following properties. The second number is 3 more
than the first number. So let's say S for second number
and F for first number. So the second number is 3 more
than the first number. So the second number is equal
to the first number plus 3. That's from statement 1. The product of the two numbers
is 9 more than the sum. So the product of the two
numbers, that's S times F is 9 more, 9 plus their sum,
plus S plus F. So let's see, we have two
equations and two unknowns. This is nonlinear because I'm multiplying these two variables. But I think we should be
able to solve them one way or the other. So let's see, we have
what S is equal to. So let's just substitute that
back into this equation. So let's say that S is
equal to F plus 3. So if we substitute for these
S's, we get F plus 3 times F is equal to 9 plus F plus 3,
right, instead of an S, F plus 3 and then plus F. Let's see if we can
simplify this. F times F is F squared plus
3F is equal to 9 plus 3 is 12 plus 2F. Subtract 2F from both sides,
you get F squared plus F is equal to 12. Subtract 12, you get F squared
plus F minus 12 is equal to 0. This one looks factorable. I don't have to take out
the quadratic equation. Let's see, this is F plus 4
times F minus 3, right? Because when you multiply those,
you get negative 12. When you add those, you get plus
1, so that is equal to 0. So in order for this to be true,
one or both of these have to be equal to 0. So if F plus 4 is equal to
0, that means F could be equal to minus 4. If F minus 3 is equal
to 0, then that says that F could be 3. So F could be minus 4 or 3. Now, S is F plus 3, so if we're
dealing with the minus 4 scenario, if F is equal to
minus 4, then what is S? Then S is going to be minus 4
plus 3, and S is going to be equal to minus 1. Then if F is equal to 3,
than S is equal to 6. So let's see if we see either
of these combinations. Minus 4, minus 1,
that's choice B. Excellent. All right, problem 34. Let me see, maybe I should
copy and paste these word problems so we can see how we
parse the problems. So I've copied it, then I'll go here,
and then pasted it. Jenny is solving the equation x
squared minus 8x equals 9 by completing the square. What number should be added to
both sides of the equation to complete the square? So x squared minus
8x is equal to 9. I wrote it with space
for a reason. When you're completing the
square, you're trying to turn the left-hand side of this
equation into some type of a perfect square. So if it's a perfect square,
I have two numbers. It's the same number that when
you add them together, you get minus 8, and when you square
them, you should get something else, right? So what's half of minus 8? Half of minus 8 is minus 4. So minus 4 squared is 16. So if I add 16 to both
sides, I'm all set. Why did that work? Well, now it's a
perfect square. This is now x minus 4 squared
is equal to 9 plus 16 is 25. They're not even asking
us to solve it. They just want to know what we
had to add to both sides. So it's 16, D. Remember the whole logic here,
and I've done a few videos on completing the squares, is what
number do I add here to make this a perfect square? You say, OK, I have a minus
8x, so I take half of this number, because the same number
added to itself twice is going to become minus 8. I take half of that number,
then I squared it. So half of minus 8 is minus 4. If you square it,
you get the 16. So add 16 to both sides,
you get this. You can actually
solve for this. x minus 4 is plus or minus 5. You keep going. That's actually where the
quadratic equation comes from. Anyway, next problem. 16 was choice number D. I'm going to copy and paste
this entire problem here. Let's go here and
paste it here. OK, which of the following most
accurately describes the translation of the graph y is
equal to x plus 3 squared minus 2 to the graph y equals
x minus 2 squared plus 2? So the y translation tends to be
pretty easy to figure out. Let me just draw some
example graph. So if I had the graph x squared,
the graph x squared looks something like this. Let's see if I can draw it. The graph x squared looks
something like this, right? And it intersects. When x is equal to 0, we're
at our minimum point. And any other value increases
in both directions. The graph of x squared plus
2, you're shifting up. This is the graph of
x squared plus 2. You would shift it up by 2. The graph of x squared minus 2,
you would shift down by 2. This would be x squared
plus 2. This would be x squared
minus 2. So the shift in the y direction
is very easy to see. So if we're going from something
minus 2 to plus 2 we're going to be shifting
it up 4, right? So that's always the
easy one to just eyeball and figure out. So we're definitely going to be
shifting from minus 2 to 2. So it's up 4. It's either going to be
choice A or choice D. The left/right shift is often
a little bit more hard for people to visualize or to at
least internalize, but I'll give you an attempt. Let's just go back to this. This is the graph of x squared,
this yellow line right there. That's the graph of x squared. Let me ask you a question. What is the graph of
x minus 3 squared? So does this shift it down 3 to
the negative direction or 3 to the positive direction? Your intuition might say,
oh, I'm subtracting 3. When I did minus 2,
I shifted down. But it's actually the
opposite here. Because you have to think about
for what value of x am I going to have a 0
squared here? That happens with
x is equal to 3. So you can think
of it this way. Now, when we're at this point,
when x is equal to 3, it's the same thing as this point when
we have just x squared. When you put 3 in here, this
whole expression becomes zero. As you get above 3, that's
like going above zero. As you go below 3, that's
like going below zero. So this graph will just get
shifted to the right by 3. That's x minus 3 shifts
to the right by 3. x plus three would go in the
other direction, because when x is minus 3, that's when
it would equal to zero. I haven't written that down. So let's think about this. We're going from x plus 3, so if
this is x squared, x plus 3 would look something-- let me do
it in a different color. x plus 3 is actually shifted
to the left. The way I always think about
it, there's two ways to think about it. The y shift is intuitive and
the x shift might not be. If you have a plus 3 here,
you're actually shifting in the downward direction. The way to actually think about
the intuition is when will this whole expression
equal 0? This whole expression equals 0
when x is equal to minus 3. So that's the point at which
you're getting 0 squared. When I'm drawing these
graphs, I'm not doing the y shift here. So this is going to be shifted
to the left 3. This is going to be shifted
to the right by 2. So if this is shifted to the
left 3 and this is shifted to the right by 2, to go from this
to this, you're shifting to the right by 5. So the actual graph x plus
3 squared minus 2 is going to be here. Then to go here, you have a plus
2, so you're shifting the graph up by 4, and then you're
going to x minus 2. So this graph right here
is going to be up here. So you're shifting up by 4 and
then you're shifting to the right by 5. Actually, even if you're
confused with your shifting left or right, you can just
say the difference between plus 3 and minus 2 is 5,
and 5 is only there. But you should hopefully
understand the problems a little bit deeper than that. Anyway, I'll see you
in the next video.