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CCSS Math: HSF.BF.A.1b

- [Instructor] We're told
that f of x is equal to two x times the square root of five minus four. And we're also told that g
of x is equal to x squared plus two x times the square
root of five minus one. And they want us to find g minus f of x. So pause this video, and see if you can work
through that on your own. So the key here is to just
realize what this notation means. G minus f of x is the same thing as g of x minus f of x. And so, again, if this was helpful to you, once again I encourage
you to pause the video. All right, now let's
work through this again. So this is going, or,
I guess the first time, but now that we know that this is equal to g of x minus f of x. So what is g of x? Well, that's the same thing
as x squared plus two x times the square root of five minus one. And what is f of x? Well, it's going to be two x times the square root of five minus four. And we are subtracting f of x from g of x. So let's subtract, this
is f of x, from g of x. And so now it's just going to be a little bit of
algebraic simplification. So this is going to be equal to, this is equal to x squared plus two x times
the square root of five minus one. And now we just have to
distribute this negative sign. So negative one times two x times the square root of five is, we're gonna have minus two x
times the square root of five. And then the negative of
negative four is positive four. Now let's see if we
can simplify this some. So this is going to be equal to, we only have one x squared term, so that's that one there. So we have x squared. Now let's see, we have two x
times the square root of five. And then we have another, oh, and then we subtract two x
times the square root of five. So these two cancel out with each other. So those cancel out. And then we have minus one plus four. So if we have negative one
and then we add four to it, we're going to have positive three. So if we just fact, if we take this and
this into consideration, four minus one is going
to be equal to three, and we're done. That's what g minus f of x is equal to, x squared plus three.