1
00:00:00,467 --> 00:00:04,000
Factor 'x' squared minus forty-nine 'y' squared

2
00:00:04,000 --> 00:00:05,968
so what's interesting here is that,

3
00:00:05,968 --> 00:00:07,802
while x squared is clearly a perfect square

4
00:00:07,802 --> 00:00:08,800
-it's the square of x

5
00:00:08,800 --> 00:00:12,000
forty-nine y squared is also a perfect square

6
00:00:12,000 --> 00:00:15,133
it's the square of seven y

7
00:00:15,133 --> 00:00:17,554
so it looks like we might have a special form

8
00:00:17,554 --> 00:00:19,933
-just thing is what happens if we take:

9
00:00:19,933 --> 00:00:24,333
a plus b times a minus b

10
00:00:24,333 --> 00:00:27,000
I'm just doing it in the general case so we can see a pattern here

11
00:00:27,000 --> 00:00:29,629
this would be: a times a

12
00:00:29,629 --> 00:00:33,200
- which would be a squared-, plus: a times negative b,

13
00:00:33,200 --> 00:00:37,709
-which would be negative ab-, plus b times a,

14
00:00:37,709 --> 00:00:41,471
-or a times b again, which would be ab-

15
00:00:41,471 --> 00:00:45,302
and then you have b times negative b, which would be minus b squared

16
00:00:45,302 --> 00:00:49,000
Now, these middle two terms cancel out: minus ab, plus ab

17
00:00:49,000 --> 00:00:53,067
and you're left with a squared minus b squared

18
00:00:53,067 --> 00:00:57,133
and that's the exact pattern we have here, we have an a squared minus a b squared

19
00:00:57,133 --> 00:01:06,804
so, in this case, a is equal to x, and b is equal to seven y.

20
00:01:06,804 --> 00:01:11,471
So we have x squared minus seven y, the whole thing, squared

21
00:01:11,471 --> 00:01:16,400
So we can expand this as a difference of squares

22
00:01:16,400 --> 00:01:19,333
or actually this thing right over here is a difference of squares

23
00:01:19,333 --> 00:01:22,965
so we can expand this, like this, so this will be equal to:

24
00:01:22,965 --> 00:01:30,047
x plus seven y, times x minus seven y

25
00:01:30,047 --> 00:01:31,881
and, once again, we're just pattern matching,

26
00:01:31,881 --> 00:01:34,000
based on this realization right over here:

27
00:01:34,000 --> 00:01:37,200
if I take a plus b, times a minus b, I get a difference of squares

28
00:01:37,200 --> 00:01:39,133
this is a difference of squares,

29
00:01:39,133 --> 00:01:43,200
so when I factor it, it must come out to something that looks like

30
00:01:43,200 --> 00:01:47,369
a plus b times a minus b, or x plus seven y, times x minus seven y