Video Transcript
Which of the following expressions
is equivalent to 16𝑥 squared minus a quarter? The options are A) 16𝑥 plus a half
multiplied by 16𝑥 minus a half, B) 𝑥 multiplied by four 𝑥 minus a half, C) four
𝑥 plus a half multiplied by four 𝑥 minus a half, or D) the square root of four 𝑥
minus a half.
So if we take a look at our
expression 16𝑥 squared minus a quarter, what this is an example of is the
difference of two squares. So the difference of two squares is
when we have a square term and then minus another square term, so for example, if we
have 𝑥 squared minus nine because 𝑥 squared is something that can be square rooted
and nine is a term that can be square rooted. So then, we can say that this can
be written as 𝑥 minus three and that’s because the square root of 𝑥 squared is 𝑥
and the square root of nine is three multiplied by 𝑥 plus three.
So therefore, what I’ve done is
multiplied 𝑥 minus three by its conjugate which is 𝑥 plus three. And I’ll show you why this
works. Well, if I distribute across these
parentheses, what I get is 𝑥 multiplied by 𝑥 which is 𝑥 squared then plus three
𝑥 because we have 𝑥 multiplied by positive three then minus three 𝑥. And then finally, we have minus
nine and that’s because we have negative three multiplied by positive three. Negative multiplied by positive is
a negative. Well, if we simplify this, we get
positive three 𝑥 minus three 𝑥. Well, this is just zero. So they cancel each other out. So we’re left with our 𝑥 squared
minus nine which is what we started with.
Okay, great, so now we can
understand what this is. We know it’s a difference of two
squares. Let’s solve the problem. So to find the difference of two
squares, what we need to do is we need to square root each of our terms. But also, when we’re doing that, we
can square root each part of each term individually.
So to start off with, we’ve got the
square root of 16. And this is gonna be four. So we’re gonna have four at the
beginning of each of our parentheses. And then, we’re gonna have the
square root of 𝑥 squared which is gonna give us 𝑥. And this would work because if we
had four 𝑥 multiplied by four 𝑥, we get 16𝑥 squared. So now we need to add the signs
into our parentheses.
So we know that one’s gonna be the
conjugate of the other. But what I’ve done is I’ve put a
positive and then a negative, could have been the other way around but as long as
they are different signs. And then finally, we have the
square root of a quarter which is equal to a half. And that’s because the square root
of a quarter is the same as the square root of one over the square root of four
which is equal to one over two or a half.
So therefore, we can say that 16𝑥
squared minus quarter can be written as four 𝑥 plus a half multiplied by four 𝑥
minus a half, which means that the correct solution that is equivalent to 16𝑥
squared minus a quarter is C, four 𝑥 plus a half multiplied by four 𝑥 minus a
half. And we could do a quick check of
that by distributing across our parentheses.
So we have four 𝑥 multiplied by
four 𝑥 which is 16𝑥 squared. Then we’re gonna have four 𝑥
multiplied by negative a half which gives us negative two 𝑥. That’s because half of four 𝑥 is
two 𝑥 and it’s negative. And then we have positive a half
multiplied by four 𝑥 which is just add two 𝑥.
And then finally, we subtract a
quarter and that’s because we have positive a half multiplied by negative a
half. Half multiplied by a half is a
quarter. Well, the negative two 𝑥 and the
positive two 𝑥 cancel each other out. So we’re left with 16𝑥 squared
minus a quarter, which is what we wanted because that’s what we started with. So we can say, “Yes, definitely
answer C is the correct solution to this question.”
