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### Course: Algebra 1>Unit 13

Lesson 8: Factoring quadratics with perfect squares

# 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?
• 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
• 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.
• When factoring your working on making it simpler correct?
• 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

Example 2:
Factor 15

Now for two Algebraic Examples

Example 3
Multiply (x+2)(x-3)
x^2-3x+2x-6

Example 4
Factor x^2-x-6
(x+?)(x-?)
(-3)(2)=6 and (-3)+(2)=-1
• monomial, binomial, trinomial : the terminology goes down to?
• "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.
• 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.
• why does ab have to be equal to -15? where did it come from?
• 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.
• 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?
• In my opinion, if the trinomial has two possible solutions and you only answer one of them, I'll mark wrong your test because the solution is not complete.
(1 vote)
• Sorry if this question is off topic but what is the number e?
• 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 :)
• Does perfect squares just mean that we have two terms that are perfect squares, or does it mean anything else also?
• x^2-64 is a perfect square because both terms are squared and can be factored to (x-8)(x+8)
• 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