In this video, I’m going to

expose you to what is maybe one of at least the top five

most useful formulas in mathematics. And if you’ve seen many of my

videos, you know that I’m not a big fan of memorizing

things. But I will recommend you

memorize it with the caveat that you also remember how to

prove it, because I don’t want you to just remember

things and not know where they came from. But with that said, let me

show you what I’m talking about: it’s the quadratic

formula. And as you might guess, it is to

solve for the roots, or the zeroes of quadratic equations. So let’s speak in very general

terms and I’ll show you some examples. So let’s say I have an equation

of the form ax squared plus bx plus

c is equal to 0. You should recognize this. This is a quadratic equation

where a, b and c are– Well, a is the coefficient on the x

squared term or the second degree term, b is the

coefficient on the x term and then c, is, you could imagine,

the coefficient on the x to the zero term, or it’s

the constant term. Now, given that you have a

general quadratic equation like this, the quadratic formula

tells us that the solutions to this equation are

x is equal to negative b plus or minus the square root of

b squared minus 4ac, all of that over 2a. And I know it seems crazy and

convoluted and hard for you to memorize right now, but as you

get a lot more practice you’ll see that it actually is a pretty

reasonable formula to stick in your brain someplace. And you might say, gee, this is

a wacky formula, where did it come from? And in the next video I’m

going to show you where it came from. But I want you to get used to

using it first. But it really just came from completing

the square on this equation right there. If you complete the square here,

you’re actually going to get this solution and that

is the quadratic formula, right there. So let’s apply it to some

problems. Let’s start off with something that we could have

factored just to verify that it’s giving us the

same answer. So let’s say we have x

squared plus 4x minus 21 is equal to 0. So in this situation– let me

do that in a different color –a is equal to 1, right? The coefficient on the

x squared term is 1. b is equal to 4, the coefficient

on the x-term. And then c is equal

to negative 21, the constant term. And let’s just plug it in the

formula, so what do we get? We get x, this tells us that

x is going to be equal to negative b. Negative b is negative 4– I put

the negative sign in front of that –negative b

plus or minus the square root of b squared. b squared is 16, right? 4 squared is 16, minus 4 times

a, which is 1, times c, which is negative 21. So we can put a 21 out there

and that negative sign will cancel out just like that with

that– Since this is the first time we’re doing it, let me

not skip too many steps. So negative 21, just so you

can see how it fit in, and then all of that over 2a. a is 1, so all of that over 2. So what does this simplify, or

hopefully it simplifies? So we get x is equal to negative

4 plus or minus the square root of– Let’s see we

have a negative times a negative, that’s going to

give us a positive. And we had 16 plus, let’s see

this is 6, 4 times 1 is 4 times 21 is 84. 16 plus 84 is 100. That’s nice. That’s a nice perfect square. All of that over 2, and so this

is going to be equal to negative 4 plus or

minus 10 over 2. We could just divide both of

these terms by 2 right now. So this is equal to negative 4

divided by 2 is negative 2 plus or minus 10 divided

by 2 is 5. So that tells us that x could be

equal to negative 2 plus 5, which is 3, or x could be equal

to negative 2 minus 5, which is negative 7. So the quadratic formula

seems to have given us an answer for this. You can verify just by

substituting back in that these do work, or you could even

just try to factor this right here. You say what two numbers when

you take their product, you get negative 21 and when you

take their sum you get positive 4? So you’d get x plus 7

times x minus 3 is equal to negative 21. Notice 7 times negative 3 is

negative 21, 7 minus 3 is positive 4. You would get x plus– sorry

it’s not negative –21 is equal to 0. There should be a 0 there. So you get x plus 7 is equal

to 0, or x minus 3 is equal to 0. X could be equal to negative

7 or x could be equal to 3. So it definitely gives us the

same answer as factoring, so you might say, hey why bother

with this crazy mess? And the reason we want to bother

with this crazy mess is it’ll also work for problems

that are hard to factor. And let’s do a couple of

those, let’s do some hard-to-factor problems

right now. So let’s scroll down to get

some fresh real estate. Let’s rewrite the formula again,

just in case we haven’t had it memorized yet. x is going

to be equal to negative b plus or minus the square root

of b squared minus 4ac, all of that over 2a. I’ll supply this to

another problem. Let’s say we have the equation

3x squared plus 6x is equal to negative 10. Well, the first thing we want

to do is get it in the form where all of our terms or on the

left-hand side, so let’s add 10 to both sides

of this equation. We get 3x squared plus the

6x plus 10 is equal to 0. And now we can use a

quadratic formula. So let’s apply it here. So a is equal to 3. That is a, this is b and

this right here is c. So the quadratic formula

tells us the solutions to this equation. The roots of this quadratic

function, I guess we could call it. x is going to be equal

to negative b. b is 6, so negative 6

plus or minus the square root of b squared. b is 6, so we get 6 squared

minus 4 times a, which is 3 times c, which is 10. Let’s stretch out the radical

little bit, all of that over 2 times a, 2 times 3. So we get x is equal to negative

6 plus or minus the square root of 36 minus– this

is interesting –minus 4 times 3 times 10. So this is minus– 4

times 3 times 10. So this is minus 120. All of that over 6. So this is interesting, you

might already realize why it’s interesting. What is this going

to simplify to? 36 minus 120 is what? That’s 84. We make this into a 10,

this will become an 11, this is a 4. It is 84, so this is going to be

equal to negative 6 plus or minus the square root of– But

not positive 84, that’s if it’s 120 minus 36. We have 36 minus 120. It’s going to be negative

84 all of that 6. So you might say, gee,

this is crazy. What a this silly quadratic

formula you’re introducing me to, Sal? It’s worthless. It just gives me a square root

of a negative number. It’s not giving me an answer. And the reason why it’s not

giving you an answer, at least an answer that you might want,

is because this will have no real solutions. In the future, we’re going to

introduce something called an imaginary number, which is a

square root of a negative number, and then we can actually

express this in terms of those numbers. So this actually does have

solutions, but they involve imaginary numbers. So this actually has no real

solutions, we’re taking the square root of a negative

number. So the b squared with the b

squared minus 4ac, if this term right here is negative,

then you’re not going to have any real solutions. And let’s verify that

for ourselves. Let’s get our graphic calculator

out and let’s graph this equation right here. So, let’s get the graphs that y

is equal to– that’s what I had there before –3x squared

plus 6x plus 10. So that’s the equation and we’re

going to see where it intersects the x-axis. Where does it equal 0? So let me graph it. Notice, this thing just comes

down and then goes back up. Its vertex is sitting here

above the x-axis and it’s upward-opening. It never intersects

the x-axis. So at no point will this

expression, will this function, equal 0. At no point will y equal

0 on this graph. So once again, the quadratic

formula seems to be working. Let’s do one more example,

you can never see enough examples here. And I want to do ones that are,

you know, maybe not so obvious to factor. So let’s say we get negative 3x

squared plus 12x plus 1 is equal to 0. Now let’s try to do it just

having the quadratic formula in our brain. So the x’s that satisfy this

equation are going to be negative b. This is b So negative b is

negative 12 plus or minus the square root of b squared, of

144, that’s b squared minus 4 times a, which is negative 3

times c, which is 1, all of that over 2 times a, over

2 times negative 3. So all of that over negative 6,

this is going to be equal to negative 12 plus or

minus the square root of– What is this? It’s a negative times a negative

so they cancel out. So I have 144 plus 12, so

that is 156, right? 144 plus 12, all of that

over negative 6. Now, I suspect we can

simplify this 156. We could maybe bring

some things out of the radical sign. So let’s attempt to do that. So let’s do a prime

factorization of 156. Sometimes, this is the hardest

part, simplifying the radical. So 156 is the same thing

as 2 times 78. 78 is the same thing

as 2 times what? That’s 2 times 39. So the square root of 156 is

equal to the square root of 2 times 2 times 39 or we could say

that’s the square root of 2 times 2 times the

square root of 39. And this, obviously, is just

going to be the square root of 4 or this is the square root

of 2 times 2 is just 2. 2 square roots of 39, if I

did that properly, let’s see, 4 times 39. Yeah, it looks like

it’s right. So this up here will simplify to

negative 12 plus or minus 2 times the square root of 39, all

of that over negative 6. Now we can divide the numerator

and the denominator maybe by 2. So this will be equal to

negative 6 plus or minus the square root of 39

over negative 3. Or we could separate these

two terms out. We could say this is equal to

negative 6 over negative 3 plus or minus the square root

of 39 over negative 3. Now, this is just a 2

right here, right? These cancel out, 6 divided

by 3 is 2, so we get 2. And now notice, if this is plus

and we use this minus sign, the plus will become

negative and the negative will become positive. But it still doesn’t

matter, right? We could say minus or plus,

that’s the same thing as plus or minus the square root

of 39 nine over 3. I think that’s about as simple

as we can get this answered. I want to make a very clear

point of what I did that last step. I did not forget about

this negative sign. I just said it doesn’t matter. It’s going to turn the positive

into the negative; it’s going to turn the negative

into the positive. Let me rewrite this. So this right here can be

rewritten as 2 plus the square root of 39 over negative 3 or 2

minus the square root of 39 over negative 3, right? That’s what the plus or minus

means, it could be this or that or both of them, really. Now in this situation, this

negative 3 will turn into 2 minus the square root

of 39 over 3, right? I’m just taking this

negative out. Here the negative and the

negative will become a positive, and you get 2

plus the square root of 39 over 3, right? A negative times a negative

is a positive. So once again, you have

2 plus or minus the square of 39 over 3. 2 plus or minus the square

root of 39 over 3 are solutions to this equation

right there. Let verify. I’m just curious what the

graph looks like. So let’s just look at it. Let me clear this. Where is the clear button? So we have negative 3 three

squared plus 12x plus 1 and let’s graph it. Let’s see where it intersects

the x-axis. It goes up there and then

back down again. So 2 plus or minus the square,

you see– The square root of 39 is going to be a little

bit more than 6, right? Because 36 is 6 squared. So it’s going be a little bit

more than 6, so this is going to be a little bit

more than 2. A little bit more than 6 divided

by 2 is a little bit more than 2. So you’re going to get one value

that’s a little bit more than 4 and then another value

that should be a little bit less than 1. And that looks like the case,

you have 1, 2, 3, 4. You have a value that’s pretty

close to 4, and then you have another value that is a little

bit– It looks close to 0 but maybe a little bit

less than that. So anyway, hopefully you found

this application of the quadratic formula helpful.

omg he skipped too much my brain cant cope up

Nice

I’m confused on the first example you didn’t multiply 2 and the variable a

i memorized the quadratic equation from the video 'don't stay in school'

Who else is staying up at 1 AM watching this and trying to get your homework done in a panic??? 😂

When can you use the Quadratic Formula?

I eonder how much this guy gets paid

On the first example how did 100 turn into 10??

use photomath

Wait for the first problem how did you get 10 from 100?

Edit: nvm he divided

Got this formula tattooed on my wrist today frig you forever math profs.

Where did he got the 10 can somebody explain me the first question please

This may be one of the hardest I've ever learned in algebra

Thanks sir

I didnt get the last equation

Would the equation at 9:11 be an inconsistent system

I feel so stupid I cant understand how to do it 😭😭😭😭

He says really clear I study at school a whole year but I don't know how to do it because my teacher talks too quickly but when I watch this video I can understand how to do about it thank you!!!

Dude the first problem is the first problem on my actual homework wth😂

thanks for teaching this, very helpful.

I still don’t understand why you have to simplify 156 at 12:15. Please explain

i laughed when i saw this

Students: Can we learn how to do taxes?

The Education System: Screams in quadratic formula

So I just got to algebra 2 and I was freaking out because I'd never even heard of a quadratic formula, but now I understand perfectly. Why am I even using a math curriculum and a teacher?? Why not just skip the middle man and just watch Kahn academy?

At 4:49, the quadratic formula gives (-7) or (3). But the correct answer is (7) and (-3). Are the negative signs supposed to be always changed on the final answer?

Try the IMPOSSIBLE WHOPPER…. much better than quadratic formula 👍🏻

Please reply fast did you have to simplify that 156 or could you have just finished solving it. And why did u simplify the 156 but not the 84 from the second equation?????

this some maths i don't understand ;-;

how is it not 36-120? you just changed both numbers around and I don't know why. Can someone reply to my comment with an answer, please? Thanks

The answer to the first example should be POSTIVE 7 and NEGATIVE 3…?

Learned more in 16 min than i did a week in algebra 1

13:25 new word he came up with?

What happens if you can't divide divide any of the last numbers by 2 and they're odd numbers

I get a different answer when I use completing the square to solve it. I get 2+ or minus the square root of 13/3

How did you get 10?

GREAT

Is there anyone here that did csec maths 2019

Throughout the years this channel has saved my butt so many times. I hope whoever makes these videos is living a prosperous and happy life 😌

X^3-X^2+15+5/X=0. How can i solve this type of equation?

does he use a mouse or like a pen

May i go to the bathroom.

Kids: "Why are you teaching us this?"

Teacher: "

Because the state told me to, now learn!"