Main content

### Course: Algebra (all content) > Unit 9

Lesson 5: Solving quadratics by factoring- Solving quadratics by factoring
- Solving quadratics by factoring
- Quadratics by factoring (intro)
- Solving quadratics by factoring: leading coefficient ≠ 1
- Quadratics by factoring
- Solving quadratics using structure
- Solve equations using structure
- Quadratic equations word problem: triangle dimensions
- Quadratic equations word problem: box dimensions
- Solving quadratic equations by factoring (old)
- Solving quadratics by factoring review

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Solving quadratic equations by factoring (old)

An old video where Sal solves a bunch of quadratic equations by using factorization methods. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

- why they call quadratic?.(101 votes)
- It's a bit strange to have such a geometric word in the middle of algebra, but quadratic means that it's about squares, here specifically that it involves expressions where you are squaring the independent variable.(187 votes)

- In the beginning of the video he mentions that in the previous video they show you something. My question is, is there a previous video? Isn't this the first one in "Multiplying and Factoring Expressions"?(22 votes)
- You will find the concept in Factoring Quadratic Expressions. Good luck.(3 votes)

- Why does Sal say( x-12)^2=0 then he says it like this: x-12=0 where is the ^2?(10 votes)
- Consider this: For (x-12)(x-12) to be equal to zero, one of those factors have to be zero.

Ie; one of them will have to be "X=1 2; X-12 = 12-12 = 0" for it to be Zero when solved.

We see that the factors that Sal arrives at are (X-12)(X-12): Both are the same, which means their product is equal to one of them being squared. Since both the factors are equal to Zero, even if one of them is removed, the resulting number on the RHS will be Zero.

This is why Sal states that they are both the same and that (x-12) = 0. Since one of them is 0, squaring still leaves us with the solution zero. It's like two people saying the same thing in a language, with different accents.

PS: Hope this makes sense. This is how I understand the concept. If I am wrong, correct me by all means! :)(1 vote)

- what if you put 0/0=(3 votes)
- Hi, good question!

0/0 is usually designated as being indeterminate (rather than undefined, with is what is designated when a different number is divided by zero). Watch this video for more info: http://www.khanacademy.org/math/arithmetic/number-properties/v/why-zero-divided-by-zero-is-undefined-indeterminate(7 votes)

- Why do you set quadratic equations equal to zero?(4 votes)
- Lots of things multiply to become a number:

21=3*7 and 21=(21/5)(1/5). This makes an infinite number of answers that equal 21.

However, only specific things multiply to equal 0. In fact, they are only things that involve a zero! (x+7)(x-3)=0 implies that one of those two factors is 0. So the next step is to solve the two factors for x:

(x+7)=0 and (x-3)=0

x+7=0 → x=-7

x-3=0 → x=3

Therefore, the final answer is you**solve for x**is:

x={-7,3} which is read as, ex is equal to the set that contains negative seven and three.

However, the question may not ask what x is. If it only asks what the**fully factored form**is, the answer is: (x+7)(x-3)=0(2 votes)

- find the value of unknown variable ,if

xy=2 and x+y=3(3 votes)- The first thing to do when you face a problem like this is to test "easy solutions". I won't go in details of my thinking method, but the answer come right into my mind.

What two simple numbers multiplied give two? one and two. I then try them in the second equation, and it fills the solution.

x=1, y=2. There might be more solutions, but they were not asked.(4 votes)

- What if the coefficient of the first term is larger than 1?(4 votes)
- As I understand it, we use the quadratic formula, where we bring a quadratic equation to the form of "ax^2 + bx + c = 0".

Then we use the values thus got for finding x, using the formula "(-b +/- sqrt(b^2 - 4ac))/2".

The formula, when typed out in this format, looks quite intimidating. But it's quite easy to understand, and Sal does an excellent job of explaining the concepts in this video:

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratic-functions-equations/x2f8bb11595b61c86:quadratic-formula-a1/v/using-the-quadratic-formula

PS: Do feel free to correct me if I'm wrong (I'm not a teacher, just a student taking advantage of the Socratic method ^_^"). Criticism is great, as long as it's constructive. :D(0 votes)

- could someone please tell me about the rules. Ex: (a-b)(a+b)(2 votes)
`(a + b)(a + b) = a^2 + 2ab + b^2`

(a - b)(a - b) = a^2 - 2ab + b^2

(a + b)(a - b) = a^2 - b^2

If you don't remember the rules, figuring them out is not very difficult. The rules of multiplication are used. E.g.

(a + b)(a - b) =

a(a - b) + b(a - b) =

(a^2 - ab) + (ab - b^2) =

a^2 - ab + ab - b^2 =

a^2 - b^2(2 votes)

- How do you factor an expression? They don't have an equal sign. Could you give an example please?(2 votes)
- a^2 + 2ab + b^2 factors into (a + b)^2. Therefore we can equate the two as:

a^2 + 2ab + b^2 = (a + b)^2(2 votes)

- how do you do the square root of a number if it's exponent is 3 and not 2?(1 vote)
- sqr(a^3) = sqr(a*a^2) = a*sqr(a), where sqr is square root(4 votes)

## Video transcript

Let's solve some quadratic
equations by factoring. So let's say I had x squared
plus 4x is equal to 21. Now your impulse might be to
try to factor out an x and somehow set that equal to 21. And that will not lead you
to good solutions. You'll probably end up doing something that's not justified. What you need to do here is
put the entire quadratic expression on one side
of the equation. We'll do it onto the
left-hand side. So let's subtract 21 from both
sides of this equation. The left-hand side then
becomes x squared plus 4x minus 21. And then the right-hand side
will be equal to 0. And the way you want
to solve this, this is a quadratic equation. We have a quadratic expression
being set equal to 0. The way you want to solve this
is you want to factor them, and say, OK, each of
those factors could then be equal to 0. So how do we factor this? Well, we saw in the last video
that we have to figure out two numbers whose product is equal
to negative 21, and whose sum is equal to 4. This would be a plus b would
have to be equal to 4. Since their product is negative,
they have to be of different signs. And so let's see, the number
that jumps out at me is 7 and 3. If I have negative 7 and
positive 3, I would get negative 4. So let's do positive
7 and negative 3. So the a and b are positive
7 and negative 3. When I take the product,
I get negative 21. When I take their sum,
I get positive 4. So I can rewrite this
equation here. I could rewrite it as x
plus 7, times x minus 3, is equal to 0. And now I can solve this by
saying, look, I have two quantities. Their product is equal to 0. That means that one or both of
them have to be equal to 0. So that means that x plus
7 is equal to 0. That's an x. Or x minus 3 is equal to 0. I could subtract 7 from both
sides of this equation. And I would get x is equal
to negative 7. And over here, I can add 3 to
both sides of this equation. And I'll get x is equal to 3. So both of these numbers are
solutions to this equation. You could try it out. If you do 7-- negative
7 squared is 49. Negative 7 times 4 is minus
28, or negative 28. And that does indeed equal 21. And I'll let you try it out
with the positive 3. Actually, let's just do it. 3 squared is 9, plus
4 times 3 is 12. 9 plus 12 is, indeed, 21. Let's do a bunch
more examples. Let's say I have x squared
plus 49 is equal to 14x. Once again, whenever you see
anything like this, get all of your terms on one side of the
equation and get a 0 on the other side. That's the best way to solve
a quadratic equation. So let's subtract 14x
from both sides. We could write this as x squared
minus 14x plus 49 is equal to 0. Obviously, 14x minus 14x is 0. This quantity minus 14x is this
quantity right there. Now we just have to think about
what two numbers, when I take their product, I'm going
to get 49, and when I take their sum, I'm going
to get negative 14. So one, they have to be the same
sign because this is a positive number right here. And they're both going to
be negative because their sum is negative. And there's something
interesting here. 49 is a perfect square. Its factors are 1, 7, and 49. So maybe 7 will work, or
even better, maybe negative 7 will work. And it does! Negative 7 times negative
7 is 49. And negative 7 plus negative
7 is negative 14. We have that pattern there,
where we have 2 times a number, and then we have
the number squared. This is a perfect square. This is equal to x minus
7, times x minus 7, is equal to 0. Don't want to forget that. Or we could write this as x
minus 7 squared is equal to 0. So this was a perfect square
of a binomial. And if x minus 7 squared is
equal to 0, take the square root of both sides. You'll get x minus
7 is equal to 0. I mean, you could say x minus
7 is 0 or x minus 7 is 0. But that'd be redundant. So we just get x minus 7 is 0. Add 7 to both sides, and you
get x is equal to 7. Only one solution there. Let's do another one in pink. Let's say we have x squared
minus 64 is equal to 0. Now this looks interesting
right here. A bell might be ringing
in your head on how to solve this. This has no x term, but
we could think of it as having an x term. We could rewrite this as x
squared plus 0x minus 64. So in this situation, we could
say, OK, what two numbers, when I multiply them,
equal 64, and when I add them equal 0? And when I take their product,
I'm getting a negative number, right? This is a times b. It's a negative number. So that must mean that they
have opposite signs. And when I add them, I get 0. That must mean that a plus minus
b is equal to 0, or that a is equal to b, that we're
dealing with the same number. We're essentially dealing with
the same number, the negatives of each other. So what can it be? Well, if we're doing the same
number and they're negatives of each other, 64 is
exactly 8 squared. But it's negative 64, so maybe
we're dealing with one negative 8, and we're dealing
with one positive 8. And if we add those two
together, we do indeed get to 0. So this will be x minus
8 times x plus 8. Now you don't always have
to go through this process I did here. You might already remember that
if I have a plus b times a minus b, that that's equal to
a squared minus b squared. So if you see something that
fits the pattern, a squared minus b squared, you could
immediately say, oh, that's going to be a plus b-- a is x,
b is 8-- times a minus b. Let's do a couple more of just
general problems. I won't tell you what type these
are going to be. Let me switch colors. It's getting monotonous. Let's say we have x squared
minus 24x plus 144 is equal to 0. Well, 144 is conspicuously
12 squared. And this is conspicuously
2 times negative 12. Or this is conspicuously
negative 12 squared. So this is negative 12
times negative 12. This is negative 12
plus negative 12. So this expression can be
rewritten as x minus 12 times x minus 12, or x minus
12 squared. We're going to set
that equal to 0. This is going to be 0 when
x minus 12 is equal to 0. You can say either of these
could be equal to 0, but they're the same thing. Add 12 to both sides of that
equation and you get x is equal to 12. And I just realized, this
problem up here, I factored it, but I didn't actually
solve the equation. So this has to be equal to 0. Let's take a step back to
this equation up here. And the only way that this thing
over here will be 0 is if either x minus 8 is
equal to 0 or x plus 8 is equal to 0. So add 8 to both
sides of this. You get x could be equal to 8. Subtract 8 from both
sides of this. You get x could also be
equal to negative 8. Let's do one more. Just to really, really get the
point drilled in your head. Let's do one more. Let's say we have 4x squared
minus 25 is equal to 0. So you might already
see the pattern. This is an a squared. This is a b squared. We have the pattern of a squared
minus b squared, where, in this case, a would
be equal to x, right? This is 2x squared. And b would be equal to 5. So if you have a squared minus b
squared, this is going to be equal to a plus b
times a minus b. In this situation, that means
that 4x squared minus 25 is going to be 2x plus 5
times 2x minus 5. And of course, that will
be equal to 0. And this will only be equal to 0
if either 2x plus 5 is equal to 0 or 2x minus 5
is equal to 0. And then we can solve
each of these. Subtract 5 from both sides. You get 2x is equal
to negative 5. Divide both sides by 2. You could get one solution
is negative 5/2. Over here, add 5
to both sides. You get 2x is equal
to positive 5. Divide both sides by 2. You get x could also be
equal to positive 5/2. So both of these satisfy
that equation up there.