Main content

### Course: Algebra 1 > Unit 13

Lesson 8: Factoring quadratics with perfect squares- Perfect square factorization intro
- Factoring quadratics: Perfect squares
- Perfect squares intro
- Factoring perfect squares
- Identifying perfect square form
- Factoring perfect squares: negative common factor
- Factoring perfect squares: missing values
- Factoring perfect squares: shared factors
- Perfect squares

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Factoring perfect squares

Sal factors 25x^2-30x+9 as (5x-3)^2 or as (-5x+3)^2. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- can't seem to solve this problem by grouping as Sal showed on the previous video. This is what I did:

25y squared - 30x + 9

25 * 9 = 225

Factors of 225:

1, 225

3, 75

5, 45

9, 25

15, 16

none of which whose sum is equal to 30.

In the last video, Sal stated that you first find the product of the first and last monomials, find the factors of that product whose sum is equal to the middle monomial, and from there you can group. I can't find factors of 225 whose sum equals -30. What am I doing wrong?(41 votes)- actually 15 * 15 is 225 , not 15*16, so we have a.b = 225 and a+b = -30

so our factors are -15 and -15 thats why he said its a perfect square, so we have 25x^2-15x-15x+9 we factor 5x(5x-3)-3(5x-3) = (5x-3)(5x-3) = (5x-3)^2(78 votes)

- Do you have a video over regular factoring of trinomials? ex: 2x squared minus 3x minus 2. Ive been looking everywhere for a video of the sort.(5 votes)
- Yes he talks about it two videos down if you're watching the playlist thing. Or you can simply visit https://www.khanacademy.org/math/algebra/quadratics/factoring_quadratics/v/u09-l2-t1-we1-factoring-special-products-1

Also he talks about it on "Factor by grouping and factoring completey"

Hope that helped! =))))(6 votes)

- When factoring your working on making it simpler correct?(2 votes)
- Factoring is the process of turning an expression into a multiplication problem.

Simplifying is the process of performing all possible operations.

So these processes actually have opposite results. To help clarify look at the following two example problems, one with the instructions factor, the other with the instructions simplify and look how each starts with the other's answer and ends with the other's question.

Example 1:

Simplify 3*5

Answer: 15

Example 2:

Factor 15

Answer 3*5

Now for two Algebraic Examples

Example 3

Multiply (x+2)(x-3)

x^2-3x+2x-6

Answer: x^2-x-6

Example 4

Factor x^2-x-6

(x+?)(x-?)

(-3)(2)=6 and (-3)+(2)=-1

Answer: (x+2)(x-3)(11 votes)

- monomial, binomial, trinomial : the terminology goes down to?(0 votes)
- "mono" = one, as in one term

"bi" = two, as in two terms

"tri" = three, as in three terms.

This video is called "Factoring perfect square trinomials" because Sal is working with equations that have three terms.(17 votes)

- Why does it call trinomial?(1 vote)
- tri means 3, so it has 3 terms. You might want to touch up on some older subjects if you don't what it means. Maybe some video about , binomials, and trinomials.(2 votes)

- why does ab have to be equal to -15? where did it come from?(2 votes)
- The pattern for a perfect square trinomial is:

a^2x^2 + 2abx + b^2

Sal is factoring 25x^2-30x+9

He uses the middle term from the pattern and from his trinomial to get: 2ab = -30

If you divide both sides by 2, you get ab = -15

Hope this helps.(8 votes)

- In the video, Sal showed us 2 possible answers to factor out the trinomial. So, if I have to answer this question like in a test or something, am I supposed to show the 2 possible answers even though they're the same or can I show one of the 2 possible answers for the question to mark right?(5 votes)
- It depends on what your teacher asks, or how the given prompt is worded. Factoring with the GCF is different, and factoring and polynomial is different.

Personally, I would ask your teacher for help!(1 vote)

- Sorry if this question is off topic but what is the number e?(2 votes)
- Don't be sorry ;D

e is a mathematical constant. It is approximately 2.71828... (it goes on forever, it's irrational) and it is the base of the natural logarithm (no need to worry about any of that stuff until Algebra 2 or later lol xD). It is used in lots of financial matters and other things that I don't really know about xD.

A link to Wikipedia (which is always extensive and unnecessary but whatever xD):

http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

Have a Happy New Year :)(7 votes)

- Does perfect squares just mean that we have two terms that are perfect squares, or does it mean anything else also?(3 votes)
- x^2-64 is a perfect square because both terms are squared and can be factored to (x-8)(x+8)(2 votes)

- How to solve quadnomials??(1 vote)
- The other method that is used that might be easier is:

Solving With Grouping:

For example, you can factor x3 + x2 – x – 1 by using grouping.

- Break up the polynomial into sets of two.

- You can go with (x3 + x2) + (–x – 1). Put the plus sign between the sets, just like when you factor trinomials.

- Find the GCF of each set and factor it out.

- The square x2 is the GCF of the first set, and (–1) is the GCF of the second set. Factoring out both of them, you get x2(x + 1) – 1(x + 1).

- Factor again as many times as you can.

- The two terms you’ve created have a GCF of (x + 1). When factored out, you get (x + 1)(x2 – 1).

However, x2 – 1 is a difference of squares and factors again as (x+1)(x-1).

This gives you a final factorization of: (x + 1)(x + 1)(x – 1), or (x + 1)2(x – 1).

(If this method doesn’t work, you may have to group the polynomial some other way. Of course, after all your effort, the polynomial may end up being prime, which is okay.

- For example, look at the polynomial x2 – 4xy + 4y2 – 16. You can group it into sets of two, and it becomes x(x – 4y) + 4(y2 – 4). This expression, however, doesn’t factor again.

You must try grouping it in some other way. In this case, if you look at the first three terms, you’ll discover a perfect-square trinomial, which factors to (x – 2y)2 – 16. Now you have a difference of squares, which factors again to [(x – 2y) – 4][(x – 2y) + 4].)

Hope this helps. If you need me to explain further, just let me know.

Cheers!!

- THEWATCHER(5 votes)

## Video transcript

Factor 25x squared
minus 30x plus 9. And we have a leading
coefficient that's not a 1, and it doesn't look like
there are any common factors. Both 25 and 30 are divisible by
5, but 9 isn't divisible by 5. We could factor
this by grouping. But if we look a little
bit more carefully here, see something interesting. 25 is a perfect square, and 25x
squared is a perfect square. It's the square of 5x. And then nine is also
a perfect square. It's the square of
3, or actually, it could be the square
of negative 3. This could also be the
square of negative 5x. Maybe, just maybe this
could be a perfect square. Let's just think
about what happens when we take the perfect square
of a binomial, especially when the coefficient on
the x term is not a 1. If we have ax plus b
squared, what will this look like when we expand
this into a trinomial? Well, this is the same
thing as ax plus b times ax plus b, which is the same
thing as ax times ax. Ax times ax is a squared x
squared plus ax times b, which is abx plus b times
ax, which is another. You You could call
it bax or abx, plus b times b, so plus b squared. This is equal to a squared
x squared plus-- these two are the same term--
2abx plus c squared. This is what happens when
you square a binomial. Now, this pattern seems
to work out pretty good. Let me rewrite our
problem right below it. We have 25x squared
minus 30x plus 9. If this is a
perfect square, then that means that the a squared
part right over here is 25. And then that means
that the b squared part-- let me do this in
a different color-- is 9. That tells us that a
could be plus or minus 5 and that b could
be plus or minus 3. Now let's see if this gels
with this middle term. For this middle
term to work out-- I'm trying to look for good
colors-- 2ab, this part right over here, needs to
be equal to negative 30. Or another way-- let me write
it over here-- 2ab needs to be equal to negative 30. Or if we divide
both sides by 2, ab needs to be equal
to negative 15. That tells us that the
product is negative. One has to be positive,
and one has to be negative. Now, lucky for us the
product of 5 and 3 is 15. If we make one of them positive
and one of them negative, we'll get up to negative 15. It looks like things
are going to work out. We could select a is
equal to positive 5, and b is equal to negative 3. Those would work out to ab
being equal to negative 15. Or we could make a is
equal to negative 5, and b is equal to positive 3. Either of these will work. If we factor this
out, this could be either a is negative--
let's do this first one. It could either be a
is 5, b is negative 3. This could either be
5x minus 3 squared. a is 5, b is negative 3. It could be that. Or you could have-- we could
switch the signs on the two terms. Or a could be negative 5,
and b could be positive 3. Or it could be negative
5x plus 3 squared. Either of these
are possible ways to factor this term out here. And you say wait, how
does this work out? How can both of these
multiply to the same thing? Well, this term, remember,
this negative 5x plus 3, we could factor
out a negative 1. So this right here is the same
thing as negative 1 times 5x minus 3, the whole
thing squared. And that's the same thing
as negative 1 squared times 5x minus 3 squared. And negative 1 squared
is clearly equal to 1. That's why this and
this are the same thing. This comes out to the same
thing as 5x minus 3 squared, which is the same thing
as that over there. Either of these are
possible answers.