1
00:00:01,667 --> 00:00:09,200
Factor 25x^2 minus 30x plus 9. So we have a leading coefficient that's not a 1.

2
00:00:09,200 --> 00:00:16,333
It doesn't look like there are any common factors both 25 and 30 are divisible by 5. But 9 isn't divisible

3
00:00:16,333 --> 00:00:23,133
by 5. So we could factor this by grouping, but if we look a little bit more carefully here. See something

4
00:00:23,133 --> 00:00:30,733
interesting. 25 is a perfect square, and so 25x^2 is a perfect square. It's the square of 5x.

5
00:00:30,733 --> 00:00:37,200
And then 9 is also a perfect square. It's the square of 3 or actually can also be -3. This could also

6
00:00:37,200 --> 00:00:43,467
be the square of -5x. So maybe, just maybe this could be a perfect square. So let's just think about

7
00:00:43,467 --> 00:00:48,959
what happens when we take the perfect square of a binomial. Especially when the coefficient on the

8
00:00:48,959 --> 00:00:57,790
x term is not a one. We have (ax +b)^2. What will this look like when we expand this into a trinomial?

9
00:00:57,790 --> 00:01:09,067
Well this is the same thing as (ax+b) times (ax+b) which is the same thing as ax times ax.

10
00:01:09,067 --> 00:01:23,133
ax times ax is the same thing is a^2 x^2 plus ax times b which is abx plus b times ax which is another

11
00:01:23,133 --> 00:01:38,667
bax or abx plus b times b. So plus b^2. So this is equal to a^2 x^2 plus these two are the same term

12
00:01:38,667 --> 00:01:48,467
plus 2abx plus b^2. So this is what happens when you square a binomial. Now this pattern seems to work

13
00:01:48,467 --> 00:01:59,067
out pretty good. Let me rewrite our problem. We have 25^2 minus 30x plus 9 so this is a perfect square.

14
00:01:59,067 --> 00:02:10,333
Then that means that the a^2 part is 25 right over here. Then that means that the b^2 part. Let me do

15
00:02:10,333 --> 00:02:20,600
this in a different color. The b^2 part is 9. So that tells us that a could be plus or minus 5. And

16
00:02:20,600 --> 00:02:28,333
that b could be plus or minus 3. Now lets see if this gels with this middle term.

17
00:02:28,333 --> 00:02:37,733
For this middle term to work out. I'm trying to look for a different color. 2ab, this part right over

18
00:02:37,733 --> 00:02:46,200
here. 2ab needs to be equal to -30. Or another way, let me write it over here. 2ab needs to be

19
00:02:46,200 --> 00:02:57,733
equal to -30 or if we divide both sides by two, ab needs to be equal to -15. So that tells us since the

20
00:02:57,733 --> 00:03:04,667
product is negative one has to be positive and one has to be negative. Now lucky for us, the product

21
00:03:04,667 --> 00:03:10,000
of 5 and 3 is 15, so if we make one of them positive and one of them negative well get up to -15. So it looks

22
00:03:10,000 --> 00:03:20,533
like things are going to work out. So we could select a is equal to positive 5 and b is equal

23
00:03:20,533 --> 00:03:31,733
to -3. Those would work out to ab equal -15. Or we could make it a is equal -5 and b is equal to 3.

24
00:03:31,733 --> 00:03:40,933
So either of these would work so if we factor this out. Let's do this first one if could be either a

25
00:03:40,933 --> 00:03:56,400
is 5, b is -3. So either be (5x -3)^2. A is five b is -3. It could be that or we could switch the signs

26
00:03:56,400 --> 00:04:07,933
on the two terms. Or a could be -5 and b could be 3. Or it could be (-5x plus 3)^2. So either of these

27
00:04:07,933 --> 00:04:15,667
are possible ways to factor this term out. You might think "How could both of these multiply to the same

28
00:04:15,667 --> 00:04:22,000
thing?" Well this term remember this -5x plus three we could factor out a -1. So this is the same thing

29
00:04:22,000 --> 00:04:32,067
as -1 times (5x -3) the whole thing squared. And that's the same thing as -1 squared times

30
00:04:32,067 --> 00:04:42,733
(5x -3)^2. And -1 squared is clearly equal to 1. So thats why this and this are the same thing. So this

31
00:04:42,733 --> 00:04:48,667
comes out to the same thing as (5x -3)^2 which is the same thing as that over there.

32
00:04:48,667 --> 99:59:59,999
So either of these are possible answers.