In this video, we're going to talk about how to factor difference of two squares. So here's the formula you need to know. Let's say you're given a problem a squared minus b squared. b squared is a square.
This is a square. And with a minus sign, we now have a difference of two squares. a squared minus b squared can be factored into a plus b and a minus b.
So all you need to do is find the square root of a squared, which is a, and then find the square root of b squared, which is b, put them in parentheses, and separate a and b with a plus and minus sign, and you'll get the answer. So let's work on an example problem. Let's say we want to factor x squared minus 36. How do we do that? The square root of x squared is x, or x to the first power. You simply take the exponent and divide it by 2. The square root of 36 is 6. One of them is going to have a plus, the other is going to have a minus.
So that's how you can factor x squared minus 36 using this formula. Now let's try another example. Let's say we want to factor x squared minus 49. The square root of x squared is going to be x. The square root of 49 is 7. And then we're going to have x plus 7 and x minus 7. Let's try another example.
y squared minus 64. Feel free to try that problem. The square root of y squared is y. The square root of 64 It's 8, so we're going to have y plus 8 and y minus 8. So that's what you need to do in order to factor a difference of two squares.
Now let's try some harder examples. Try these problems. Let's say we have 4x squared minus 25. And also 9y squared minus 100. And then 36x squared minus 121. And finally 49x squared minus 169y squared.
So feel free to pause the video and work on these problems. Let's start with the first one. What is the square root of 4x squared?
Well, let's begin with 4. The square root of 4 is 2. The square root of x squared is x. So the square root of 4x squared is going to be 2x. The square root of 25, that's going to be 5. So the answer is 2x plus 5 times 2x minus 5. Now, let's move on to the next example. So what is the square root of 9y squared? The square root of 9 is 3, and the square root of y squared is y.
So the square root of 9y squared is going to be 3y. Now what about the square root of 100? We know the square root of 100 is 10. 10 times 10 is 100. So our answer is going to be 3y plus 10 times 3y minus 10. Now, let's move on to the next one. How can we factor 36x squared minus 121?
Well, let's find the square root. The square root of 36 is 6. The square root of x squared is x. And so the square root of 36x squared is going to be 6x. The square root of 121, that's 11. So this is going to be 6x plus 11. times 6x minus 11. For the last one, the square root of 49 is 7. The square root of x squared is x.
So we have 7x. The square root of 169, that's going to be 13. The square root of y squared is y. So we're going to have 7x plus 13y times 7x minus 13y. So that's the process that you can follow in order to factor a problem. that contains a difference of two squares.
Try these two problems 3x squared minus 27 and also 5y squared minus 80. Now notice that If we take the square root of 3, we're not going to get a nice whole number. Whenever you encounter a situation like this, focus on finding the GCF, the greatest common factor. Notice that the greatest common factor between 3 and 27 is 3. So let's take out the GCF first.
3x squared divided by 3 is x squared. 27 divided by 3 is 9. So notice at this point, we now have a difference of two perfect squares. We could take the square root of x squared and 9. So let's go ahead and factor x squared minus 9. The square root of x squared is going to be x.
The square root of 9 is 3. So this is going to be plus, that's going to be minus. Thus our answer is 3 times x plus 3 times x minus 3. Now the next one is very similar. We can't take the square root of 5 and get a whole number. It'll give us an irrational number.
So what we need to do is factor out the GCF, which in this case is going to be 5. 5y squared divided by 5 is y squared. Negative 80 divided by 5, that's going to be negative 16. So now we have a difference of two perfect squares. The square root of y squared is going to be y.
The square root of 16 is 4. And so we're going to have y plus 4 times y minus 4 with a 5 in front. So if you can't take the square root, focus on factoring out the GCF first, and then factor the difference of the two squares. Go ahead and try these two problems.
x to the fourth minus 81, and also 16x to the fourth. minus 81y to the fourth. So here we have a difference of two squares. The square root of x to the fourth, what you need to do is divide the exponent by two, you're going to get x squared. x squared times x squared is x to the fourth.
The square root of 81 is 9. So right now we have x squared plus 9. And we have x squared minus 9. Now we can't factor x squared plus 9. If we try to, we're going to get complex numbers. So let's leave that alone. But we can factor x squared minus 9. We have a difference of two squares for that part.
So let's rewrite x squared plus 9. And let's factor x squared minus 9. The square root of x squared is x. The square root of 9 is going to be 3. So we get x plus 3 times x minus 3. So this is the final answer. for this problem. That's how you could factor x to the 4th minus 81. Now let's try the next one. The square root of 16 is 4. The square root of x to the 4th, divide the exponent by 2, we get x squared.
So the square root of 16x to the 4th is 4x squared. Now the square root of 81 is 9. The square root of y to the 4th is going to be y squared. So we get 9y squared. This is plus, the other one is going to be minus. Now this one we could factor again.
Because we have another difference of two squares there. So the square root of 4x squared, that's going to be 2x. The square root of 9y squared is 3y.
So we have 2x plus 3y and 2x minus 3y. times 4x squared plus 9y squared. So that's how we could factor that particular problem. Now let's move on to the next problem. 144x to the 6th minus 225. y to the fourth.
Go ahead and try that one. In addition to that, try this one as well. 64x to the fourth minus 625y to the eighth.
So let's start with the first one. What's the square root of 144x to the sixth? The square root of 144 is 12. And the square root of x to the 6th, well we need to divide 6 by 2, so we'll get 3. This is going to be x cubed.
So we have 12x cubed. Now for the next one, the square root of 225, that's 15. 15 times 15 is 225. The square root of y to the 4th is going to be y squared. So we have 15y squared. Here we have a plus and here we have a minus.
Now we can't really take the square root of x cubed because this is an odd number. Dividing it by 2 will not give us a whole number. So we can't use the difference of squares technique again. However, notice that 12 and 15 they have a GCF.
We can pull out a 3. So if we factor out a 3 from this part, we'll be left with 4x cubed plus 5y squared. And if we take out another 3 from the second factor, we'll have 3 times 4x cubed minus 5y squared. We can multiply these two and put a 9. So our answer is going to be 9 times 4x cubed plus 5y squared.
And then times 4x cubed minus 5y squared. So that's the answer for that one. Now let's move on to the next one.
So what is the square root of 64? The square root of 64 is 8. The square root of x to the 4th, that's going to be x squared. So that gives us 8x squared. Now, the square root of 625, I'm going to use a calculator for that, that is 25. And the square root of y to the 8th, if we divide this by 2, we're going to get y to the 4th.
So 25y to the 4th. This is going to be plus, and this is going to be minus. Now... We can't really take the square root of 8 and get a nice number. So this is the end of the problem.
Now let's move on to the next type of problem. Try this one. Let's say we have 4x squared y to the 6th minus 25. How would you factor that one? So first let's start with the first term.
The square root of 4 is 2. The square root of x squared is x. The square root of y to the sixth, if we divide that by two, we get y cubed. So this becomes 2xy cubed.
The square root of 25 is 5. So it's 2xy cubed plus 5 and 2xy cubed minus 5. So that's it for that example. Now, what about this one? Let's say if we have 3x plus 5y squared minus 4x plus 7y squared. How can we factor it?
So I'm going to use brackets this time. So the square root of this term, all we got to do to find it is just get rid of the square. And that is 3x plus 5y. The square root of this term is simply what we have inside the parentheses. 4x plus 7y.
All we got to do is get rid of this square. On one side, we're going to have a plus. On the other side, a minus. Now, what we need to do at this point is we need to simplify. 3x plus 4x, that's going to be 7x.
5y plus 7y, that is 12y. Here we have 3x minus 4x, so that's negative x. 5y minus 7y. It's negative 2y. Now what we can do here is factor out a negative 1. So we can put that in the front.
So we have negative 1, and then it becomes x plus 2y, and then we can rewrite this times 7x plus 12y. So that's what we can do if you ever get a problem that looks like that. Now, let's try this problem.
36 minus x plus y squared. Go ahead and work on that. The square root of 36 is 6, and the square root of x plus y squared, all we need to do is get rid of the square, and it will be simply x plus y. The first one will be positive, the second will be negative.
You can switch it around too, but as long as you have 1 plus 1 minus, you're okay. So now let's simplify. This is going to be 6 plus x plus y, and the other one is simply 6 minus x minus y. Don't forget to distribute the negative sign to the y.
So you can leave your answer in this form, or you can leave it like this. Both answers are correct. Now let's try another similar problem. x plus 3 squared minus 16. Go ahead and try that. So the square root of x plus 3 squared is simply x plus 3. The square root of 16 is 4. And we're going to have a plus and a minus.
So now let's simplify. We have an x, and then we can combine 3 plus 4, which is 7. Here we have 3 minus 4, that's going to be negative 1. So this becomes x minus 1. So the final answer is x plus 7 times x minus 1. Now, what if you have a difference of two squares, but in fraction form? What would you do if you saw a problem like this?
Well, we could follow the same process. The square root of 4 is 2. The square root of a to the 6. Divide the exponent by 2, so it's a cubed. And the square root of 9 is 3. The square root of 49 is 7. The square root of b squared is b. The square root of c to the 8, take half of that, that's c to the 4th.
That is half of the exponent. And the square root of 121 is 11. So we have plus and minus. So that's basically it for this video.
So now you know how to factor a difference of two squares in a variety of formats that might be given to you on a test.