Algebra (all content)
- Perfect square factorization intro
- Factoring quadratics: Perfect squares
- Perfect squares intro
- Factoring perfect squares
- Identifying perfect square form
- Factoring higher-degree polynomials: Common factor
- Factoring perfect squares: negative common factor
- Factoring perfect squares: missing values
- Factoring perfect squares: shared factors
- 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:
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?(37 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(68 votes)
- at0:54how did he get the formula (ax+b) squared? How do you know when or when nt to use that formula?(17 votes)
- He wants to show us what would happen if we did (ax + b)^2. He could have really used anything though. But the reasons that he chose this one are...
1) He knows that if he squares a binomial, he will get a trinomial (which is what he has in the video). So if he wants the most typical binomial squared, he would use (a+b)^2
2) But since the leading coefficient (the coefficient of the x^2) is not 1, he doesn't want to use (a+b)^2. Instead (ax + b)^2 would work better for this situation.
So, if you want to know when to use (ax+b)^2, here is the answer:
If you want to factor a second degree trinomial as a perfect square that doesn't have 1 as the leading coefficient.
Hope this helps!(6 votes)
- Why does it call trinomial?(2 votes)
- 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.(3 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.(6 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! =))))(2 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.(14 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.
Now for two Algebraic Examples
(-3)(2)=6 and (-3)+(2)=-1
Answer: (x+2)(x-3)(6 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?(4 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)
- 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)(0 votes)
- How do you factor something like 4x^2+13x+9 or is it already fully factored?(3 votes)
- So you could say that to tell quickly whether or not any trinomial is a true perfect square, you have to make sure the square roots of the first and last terms sum up to the coefficient on the middle term?(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):
Have a Happy New Year :)(3 votes)
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.