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Example 1: Factoring quadratic expressions

Factoring trinomials with a leading 1 coefficient

Example 1: Factoring quadratic expressions

Discussion and questions for this video
What happens at 3:35???? I got really confused at that part.
He factors (t+3) out. You can treat t*(t + 3) + 5*(t+3) as though it has parenthesis around it (t*(t + 3) + 5*(t+3)). If you do that you can see that you have two things (t and 5) multiplied by (t+3) so you can factor out a (t+3). This leaves you with a (t+3) multiplied by what is left inside the parenthesis which is (t+5). So the expression ends up being (t+3)(t+5)
where is the other t+3 gone can anyone explain please
You didn't specify at which time but I guess that you meant somewhere at 3:30 where we have:
(t²+3t) + (5t+15)
t(t+3) + 5(t+3) → We factored out t from (t²+3t) and 5 from (5t+15)
(t+3) (t+5) → now this is part that you were asking for, you're wondering where did the other (t+3) go, actually if you look at t(t+3) + 5(t+3) carefully you might realize that they both have (t+3) so we can just factor that out and add their coefficients in another parenthesis as another variable.
And that is why (t+3) (t+5) is our final answer!
When doing the KhanAcademy lessons for Factoring Polynomials 1 (https://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_factoring/e/factoring_polynomials_1) I was given this problem to factor.

x2−10x+24 (x to the second power minus by ten x plus twenty four)

I wrote down this answer and it solves the right way,

(x + 2)(x - 12)

But KhanAcademy said that I got it wrong and that the answer was actually,


Can anybody tell me where I went wrong? I just don't get it. They both solve the same way.
So the equation they want is x^2-10x+24 Correct?

If you multiplied yours out, it would be x^2-10x-24.
(x+2)(x-12) = x*x-12x+2x-24 = x^2-10x-24 because 2 times -12 is -24.

While the answer provided
(x-6)(x-4) = x*x-6x-4x+24 = x^2-10x+24. Negative 6 times negative 4 is positive 24.Which is what they wanted.

You got really close, just gotta watch that minus sign on the last term.
Why do the (t+3)'s you factored out of the trinomial cancel each other, and if they did wouldn't they leave a 1 instead of a a single (t+3) remaining? sorry if this was a badly worded question btw
what if the problem is x^2-2x-8. the factors of 8 dont add up to 2? like in the video 5 and3 add up to 8
-4*2 = -8
and -4+2 = -2
So x²-2x-8 can be factored.

There are other methods such as "completing the square" and the quadratic formula that can be used when you can't easily factor the expression.
Here is a video on completing the square

And on the quadratic formula https://www.khanacademy.org/math/algebra/quadratics/quadratic_formula/v/using-the-quadratic-formula

I hope that helps make it click for you.
I want to see an example of factoring a trinomial in the type of ax(squared) + bx + c where a is greater than 1 and a, b, c do not have a common factor greater than 1.
This is close to what you wanted: http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_formula_tutorial/v/using-the-quadratic-formula

And the practice session here :http://www.khanacademy.org/math/trigonometry/polynomial_and_rational/quad_formula_tutorial/e/quadratic_equation
has some problems that fit your request. If you use the "i'd like a hint" button, it will show you how to work each problem step by step.
I have a question?



Thanks, Please help!
3/2x + 4 = -9. To solve for x, we can do a few things. I would suggest one method though, let me show you...
3/2x + 4 = -9
subtract 4 from both sides.
3/2x = -13
Now, multiply both sides by 2/3.
(2/3)*3/2x = (2/3)*(-13/1)
6/6x = -26/3
6/6 = 1, so
x = -26/3
and we are done. Tell me if that helped or not.
at 2:27 he mentions grouping, what is grouping
Taking out separate thing and putting them into groups
2xto the forth +2xto the third-xsquared-x
Sal is an awesome teacher explaining it in short times and no questions needed for his methods
What if at 3:40 you had t(t+3) + 5(t-3). How do you deal with one of the values being negative?
Is there an easier way of doing this?
You find factors for the last term and then add them together and the ones that add together to equal the middle coefficient are the ones that you put after x.

Eg. t^2+8t+15
The factors of 15 are 5 and 3, and 1 and 15. 5 and 3 add together to make 8, which is the middle coefficient, so the factored equation is (x+3)(x+5)
After watching this video, I still cant figure out how to solve my problem. t squared plus 7t+12.
Sometimes you may not be able to group them easily and that is why there are other methods. BUT this one is easy and it can be solved by grouping method.
t^2 + 7t + 12 =0
=> t^2 + 4t + 3t + 12 =0
=>t(t+4)+3(t+4) = 0
=>(t+4)(t + 3) = 0
=> t+4 = 0 OR t+3 = 0
=> t = - 4 or t = - 3
Do watch the other videos by sal about solving quadratic equations.
I hope this helps !
so the final answer, the thing you would write down and box is "(t+3)(t+5)" right?
Yes; all you're doing when you're factoring a polynomial is breaking it down into its respective binomials, or the terms you multiplied together to create the polynomial.
Please Help! How would I factor the quadratic 6x^2-45x-24=0, thanks.
First lets notice that all of the coefficients are divisible by 3. We can factor that out to get 3(2x^2-15x-8).
So now we have ax^2 + bx + c and we're looking for something with the form of (mx + p)(nx + q) where a = mn, b = pn + mq, and c = pq.
a = 2 so m and n have to be 1 and 2. We now have 3(2x+p)(x+q).
b = p + 2q and c = pq. c also equals -8 so p and q must be { -1, 1, -2, 2, -4, 4, -8, 8}. If you test out all of the option you'll see that 8 * -1 = -8 (c) and -1 + 2 * 8 = -15 (b).
Our answer is 3(2x - 1)(x + 8)
(x - 1)(x + 3) = 4 what is the next step
Expand the polynomial, to get x^2 + 2x - 3 = 4
Set the equation equal to zero, and then factor.
Do you have videos that show you how to factor this type of polynomial ---> x3-8 or 10x3-25
Is there a simpler way to do this?
Well, to be fair to teachers, they do have questions asked all the time. . . . but as for the 'taught it better' part, that depends on the teacher. A bit of a redundant statement, even by my standards. And the awesome thing is that we can ask our questions in the comments and know that we won't be shunned by the world for asking a redundant question, or a simple question that many people know but a few just can't grasp.
what if you have a problem such as 9x^2+72x+142=0 where none of the numbers you mutiply together to get 142 and have a sum of 72
What if the leading coefficient is -1?
I'm starting to get the concept of the factorizing idea. But what if I had to imply on harder polynomials, for example this one in my book:
The answer is (9p^2-q^2)^2

Note that you can treat the p^4 as (p^2)^2 and q^4 as (q^2)^2.

It may be simpler to look at it this this way.
Make the definition of
p^2 = a
and the definition of
q^2 = b.

Then the original expression can be written as 81a^2 - 18ab + b^2.

This factors into (9a - b)(9a - b).

Then make the substitution a = p^2 and b = q^2. This will give you
(9p^2 - q^2)(9p^2 - q^2)
(9p^2 - q^2)^2
ok, what if there is a variable in the middle like X to the second,minus x, minus 20?
You can still factor x2-x-20. the -x would be the same as -1x. I prefer plugging these things into the quadratic equation for then you get exact answers without the guess and check method.
What is a coefficient?
A coefficient is the constant number in front of a variable that is multiplying by it. So if you have "5x", 5 is the coefficient.
My teacher in the past two years dont help and i been watching you for one hour and i alraedy get it? I need u as my teacher
Thanks... This was really helpful.
say the sum is 4 x^3 y^3 - 12 x^4 y^5 + 16 x^6 y^6
for the multiplication and addition part im unable to get the right numbers
please help me out
what's the advantage of factoring. If you simplify you go the other way right? Remove parentheses. Just wondering.
How do you factor a problem such as m^2+m-20 ?
m^2 -5m + 4m -20 =0
ok, so here goes nothing! so we know that we have to find a value for a and b. a times b should be equal to 15 and their sum should be equal to 8. so a and b are 5 and 3. this makes the equation: t(t+3)+5(t+3). so since there are two (t+3)'s then we can factor it out to this: (t+3)(t+5) and that would be the answer! does that help? hope ie does!
And what about this? Can you factor this out by grouping?
Cause i always see examples that i can solve very quickly, but i haven't see an example what i can't factor by grouping. Is that possible or i just can't see the factors in my example?
My question for my my friend, is where does your name come from? Ive never heard it before and it is interesting...
thank u.u have helped me very much.can you do simultaneous equations please?
I am working on Factoring Trinomials using FOIL. However, I am having a problem finding a video to explain that. Can you point me in the right direction? Thanks!
I am sorry, but Sal doesn't have a video on this topic. I found someone else who does a really good job on this. http://www.youtube.com/watch?v=uFenBVfPxio
What's coefficient?
The number in front of the variable. An example is 9x where 9 is the coefficient and x is the variable.
why not just use the xbox method?
I need some homework help.
The problem is 20t^2 - 12t - 32
I know the answer is 4(5t-8)(t+1) but I don't understand how my teacher got to that answer.
x^2+xz/3: use x=3, and z=5
I understand how to factor these equations but I don't understand when the leading coefficient is greater than 1. For example, a problem I received in class was 4x^2+12x-7. How would this be solved? Someone please help me. Thanks.
how would you factorise 2x squared +3x -7 = 0
Perhaps that quadratic equation cannot be factored. Try using the Quadratic Formula instead and see what kind of solutions you get for x. If they involve irrational numbers, then the equation is "prime" and cannot be factored.
Is 6x^2-13x+5 = (3x-5)(2x+1)?
No. Use FOIL First Inside Outside Last to do it backwards

3x(2x) -10x +3x - 5 =
6x^2 -7x -5
What do you do when it's 2x to the 2nd + 8t + 15?
What would I do if I had a c that couldn't be divided by my b?
Like in b^2+*b=7?
Why is B there in (t+b)?
A trinomial follows the form a^2+2ab+b^2, where a and b do not equal 0. When factored, the trinomial equals (x+a)(x+b). A and b are the variables commonly used to write out the formula for factoring a trinomial, but really any two symbols can be used.
how do i get the coefficient, and the numbers,variables, to have a similar factor?
need a more clearer question, what the topic? You can multiply an entire equation by some factor to match up other factors, say in elimination.

x^2+4x+5 can be multiplied by say 2 to get 2[x^2+4x+5] = 2x^2+8x+10 or by whatever you want to change the coeffients to.
Ok, so I have a question. If i had a problem like this: x^2+x-90, what would i do? because the middle term only has a variable x. Can someone help please?
Ok! I got it! Thanks!
I'm only confused about problems where "a" (in ax^2 + bx + c) does not have a value of 1. How do you take this into account when factoring?

Without using the quadratic formula.
If you have ax^2 + bx + c = 0 and you want to factor, the first thing I would do is divide both sides by a. Here is an example: 4x^2 + 12x + 9 = 0. Dividing both sides by 4 yields: x^2 + 3x + (9/4) = 0. You might be able to recognize this as a perfect square: The square root of 9/4 is 3/2 and (3/2)x + (3/2)x = (6/2)x = 3x. So we have x^2+3x+(9/4) = (x + 3/2)^2
I would like to know what video would inform me on how to do a problem like y=(x+9)². It is a quadratic equation.
these vids seem a little backwards in their order
oops, sorry only just realised they were different :-/
Is there any way this system of factoring can be incorrect?
I don't get from 3:35 to the end. how did he factor it out to where from t(t+3) + 5(t+3) to the answer being (t+3)(t+5) ? Please Help!
okay this follows the distributive property, he distributed the t+3 out because
(t+3)(t+5) can be rewritten as t(t+3) + 5(t+3) if you distribute the t+3...did that make sense? its like when you have 15+30 and you can make it (5)(3)+(5)(6) = (5)(3+6)
is the a and b step really necessary i feel that there is a easier way to factoring if so please tell me
The video is showing more detail of how factoring works.
When you are solving problems with a first coefficient of 1 (on the x^2), just look for factors that have:
sum = coefficient of x term
product = constant term
`Factor: x^2 + 6x + 8`
you need 2 numbers with a sum of 6 and a product of 8
The numbers will be 4 and 2
`(x+4)(x+2) will be the answer.`

Harder example:
`Factor: x^2 + 5x - 6`
You need two numbers with a sum of 5 and product of -6
The numbers are 6 and -1
`(x+6)(x-1) is the answer`
i can't use what sal is doing in the problem in my homework..

There are a couple of ways to approach this, but factoring always starts by:

1) Look for a GCF (greatest common factor)
2a^2-5a-16 There are no common factors.

2) Look for a common pattern (a^2-b^2)=(a+b)(a-b); a^2+2ab+b^2=(a+b)^2; a^2-2ab+b^2=(a-b)^2
These are three patterns you just have to brute force memorize called difference of squares and perfect square patterns.
None of these patterns work.

3) Use a generic factoring method (ac method, slide and divide, grouping, educated guess and check, etc.)
ac method: 2•-16=-32, so look for factor pairs that multiply to equal -32 and add to equal -5
There are no other factor pairs of -32 so this will not factor evenly (aka it is 'prime')

4) For more advanced factoring of quadratics you can use the quadratic formula to get irrational and imaginary factors. a*x^2+b*x+c will factor as a(x-r1)(x-r2) where r1=(-b+√(b^2-4ac))/(2a) and r2=(-b-√(b^2-4ac))/(2a)


why don't just do the magic x method?
It is kind of a homework question, practice question. I have worked on it for two days!! x(4x-5) =114. I cannot find anything that factors out. 4x-5x-114=0 What adds to 5 and when multiplies gives you 114??
Hi Patty,
Nikola is correct in the factoring but I wanted to point out where I think you were having trouble. It sounds like you were forgetting about the 4 in front of the x^2 when you were looking for factors. Remember that for Ax^2 + Bx + C = 0, when we are looking to break down the expression, we multiply A*C to get a value. In this case, A*C = -456. We then break down the -456 into two factors that, when added together, give us B, in this case, -24 and 19 are the two factors we want. We then rewrite the original equation, using our two factors to replace B:
4x^2 - 24x + 19x - 114 = 0
We can then group and factor:
(4x^2 - 24x) + (19x - 114) = 0
From the first group we can factor out a 4x and from the second group we can factor out a 19:
4x(x - 6) + 19(x - 6) = 0
At this point, since we have like terms (x-6), we can combine the 4x and the 19:
(4x + 19)(x - 6) = 0
From there, if you are solving for x, you would set each factor equal to zero:
x - 6 = 0
x = 6
4x + 19 = 0
4x = -19
x = -19/4
So your roots would be x = -19/4 or x = 6
Hope that helps.
I need help determining when to factor and when to apply the quadratic equation. For instance, in the problem:
Solve for x:
My first inclination was to apply the quadratic formula, which gave me x= (-3+or- 2(70)^1/2)/2
And that's OK, but not pretty.
But when I entered it, it was wrong. So I tried again and got the same answer. So I asked for a hint and they started in factoring. And I ran with the whole factoring thing and got the nice, pretty answer x=-10 or x=7.
Why did the two methods get me different answers?
And how can I tell when looking at a problem which one to use?
I understand everything except at 3:28, when t(t+3) + 5(t+3) equals (t+3)(t+5), I do not know how Sal got this, plz someone explain to me....

By the way Sal is awesome!!! LOL
At 3:45, how does Mr Khan go from (tsq.+3t)+(5t+15) to (t+3)*(t+5)?
What do I do if the exponent with the first monomial is a 3?
Ex: 6x^3-13x^2-28x
For that problem in particular, you can factor out an x from all the terms so that you have x(6x^2-13x-28), and doing the factoring we've been learning in the last few videos, we get x(3x+4)(2x-7).
How does he get from t(t + 3) + 5(t + 3) to (t + 3)(t + 5)

I'm interested why and how the factoring out of t + 3 works, exactly which places are factored out.
Answering my own question with the help of my GF, important is the following math rule:
a * b + a * c = a (b + c)

*Why is this?*
(a * b) + (a * c)
____________ * a = (b + c) * a

So to get from t(t + 3) + 5( + 3) to (t + 3)(t + 5) we can do the following:

t(t+3) + 5(t+3)
____________ * (t+3) = (t+5) * (t+3)
how do you solve 4x2 - 6x - 30 ???
I've tried factoring it but can't seenm to find an answer
Since the coefficient of x^2 is 4. You have to solve this:
a + b = -6
a.b = -120 [the coefficient multiplied by -30]
At 4:00, is that the answer for this problem? I still don't get it how does that when mutiplied do we get t squared? Is it the the one in purple with green or the one you were talking about t +3 times t+5. I still don't understand.
Okay, so i understand this, but what if the second number in the trinomial was posotive and the third one was negative? How would you solve the problem x squared +14x-49???
The example you gave isn't factorable with this method. However in general if the second term is positive but the third is negative, then that means the factored expressions will be (t+a)(t-b), ie. In the brackets one will have a positive in it while one will have a negative.
what is the factor trinomial for y2+y-20
Hi everyone, for factoring quadratic expressions how do you know what numbers to put in the parenthesis. I understand that you find numbers that multiply to whatever x and add up to the other number but does anyone have tips for finding out these numbers??
You can factor the last number (15 in the example) to see which of its factors add up to the middle number (8 in this case). The factors of 15 are {15,1} and {5,3}. Since 5 +3 = 8, you can try them to see if they work.
(t + 5)(t+3).= (t^2 + 8t +15)
Also, since the middle number was positive, we didn't need to consider the negative factors of 15, namely {-15,-1} and {-5,-3}. If the middle number was negative (-8) then {-5,-3} would have been the answer since
(t - 5)(t - 3).= (t^2 - 8t +15)
Does the method shown in the Factoring quadratic expression video only work if the term with an exponent is squared? Could it work if it was an equation with an exponent of 4?
Am I performing a correct arithmetic operation if I factor like following:
I haven't seen anything like it, so I got a bit confused.
It is an expression that has two or more terms, in this case there are 3 terms.
At 1:58, Sal says you can use this method if the coefficient is 1.
What if the coefficient of x term raised to the second power, is greater than one?
Like: 2x^2 + 2x + 1
Can you use this same factoring method?
You can do something similar.
2X^2 +10X + 8.
Multiply A and C. 2*8 = 16.
Find factors of AC (16) that add to equal B (10). These are 8 and 2.
Divide each factor by A. 8/2 and 2/2.
Have each coefficient of X as 1
(X+ )(X+ )
and put your factors in
Multiply this by A
2(X+8/2)(X+2/2). This reduces to
The same thing that happens when A is 1, but multiplying and dividing by 1 don't change your values.
Can x^2+24+143 be factored, having trouble with the operation.
You can save some time by noticing that you only have to check for divisibility up to the square root of the number. For example, when checking factors of 143, we only have to go up to 11 because sqrt(143)~=11.9. All of the factors of 143 that are above 11.9 will be flushed out when you test the lower numbers.

In other words, you can stop checking factors when the result you get is smaller then the number you divided by.
Whats going on? Where did Sal get a and b from? can someone explain this to me?
The a and b are what he uses to represent the numbers and can be used to represent the the coefficients of terms in any quadratic expression
How do you do the problem a^3-9a+3a^2-27? How do you factor it. I'm really confused
Continuing from Kate Ballard's explanation, the first quantity can be factored more using the difference of two squares. The final answer should be (a-3)(a+3)(a+3) and this can be changed to (a-3)(a+3)^2.
I was factoring polynomials 1 and the problem was x2(exponent)+3x-54 and I got +9 and -6 and it said it was wrong what did I do wrong?

You probably factored the polynomial correctly into (x+9)(x-6). But remember, if you were trying to solve for x, the equation is x^2+3x-54=0. Don't forget about the "equals 0" part.

So what does that mean? Well we get everything factored into:
(x+9)(x-6)=0 That means that we have 2 quantities multiplied together that equal 0, so either the first one is 0, or the second one is. So we have two possibilities, which is why we get two answers.

First possibility: x+9=0 so x=-9
Second possibility x-6=0, so x=6.

Does that sound about right?
A good shortcut would be finding the factors of the last monomial or variable which you can add or subtract and the answer would be the numeral coefficient in the middle.
That sounded more complicated
I think he was just showing the basis of the concept.
Yeah, the way you explained was how I was taught too.
Sal is factoring whole numbers. What about fractions? Ex: x^2 - 4/3x + 4/9
The factors of 4/9 are: 1, 4/9 or 2/3,2/3.
because in the original expression the middle term is negative the 4/9 is positive we are looking for two negative factors that add up to -4/3.

how do I factor by grouping polynomials to the third power? ie.
12n to the third -28n squared -15n + 35?
You factor by grouping.
You essentially create two binomials through grouping, so you have:
(12n^3-28n^2) and (-15n+35)
Then, you reduce each binomial by taking out a term (i.e. dividing by 5 or 5n or something) so the binomials left int the parentheses are equivalent:
4n^2(3n-7) -5(3n-7)
So in the equation above, 4n^2 was taken out of the first binomial by dividing, and -5 was taken out of the second binomial through division as well. Then, you can reduce this even further by taking out, or factoring out the 3n-7 and leaving 4n^2-5:
The above is your answer.
sigh... I'm so terrible at grasping concepts... I will give you a problem i dont understand..
If i factor by grouping: 7w squared+14w+wb+2b... how does it turn out to be (w+2)(7w+b)????
(w+2)(7w+b)= w times 7w which gives you 7w squared plus w times b which gives you wb plus 2 timea 7w plus 2 times b.this gives you 7w squared+wb+14w+2b.when u put them in order from greatest to least you get 7w squared+14w+wb+2b.Hope this is helpful!
Report a mistake in the video

At 2:33, Sal said "single bonds" but meant "covalent bonds."

Report a mistake in the video

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