Video: SAT Practice Test 1 • Section 3 • Question 15

Which of the following expressions is equivalent to 16𝑥² − (1/4)? [A] (16𝑥 + 1/2) (16𝑥 − (1/2)) [B] 𝑥(4𝑥 − (1/2)) [C] (4𝑥 + 1/2)(4𝑥 − (1/2)) [D] √(4𝑥 − (1/2)).

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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.”

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