I think we’ve had some pretty

good exposure to the quadratic formula, but just in case you

haven’t memorized it yet, let me write it down again. So let’s say we have a quadratic

equation of the form, ax squared plus bx,

plus c is equal to 0. The quadratic formula, which we

proved in the last video, says 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. Now, in this video, rather than

just giving a bunch of examples of substituting in the

a’s, the b’s, and the c’s, I want to talk a little bit

about this part of the quadratic formula, this

part right there. The b squared minus 4ac. And we’ve seen it in a couple of

the problems we’ve done as examples, that this kind of

determines what our solution is going to look like. If, for example, b squared minus

4ac is greater than 0, we’re going to have two

solutions, right? The square root of some

positive number that’s non-zero, there’s going to be

a positive and negative version of it– we’re always

going to have a b over 2a or negative b over 2a– so you’re

going to have negative b plus that positive square root, and

a negative b minus that positive square root,

all over 2a. So if the discriminant is

greater 0, then that tells us that we have two solutions. Now I just used a word, and

that word is discriminant. And all that is referring

to is this part of the quadratic formula. That right there– let me do it

in a different color– this right here is the discriminant

of the quadratic equation right here. And you just have to remember,

it’s the part that’s under the radical sign of the

quadratic formula. And that’s why it matters,

because if this is greater than 0, you’re having a positive

square root, and you’ll have the positive and

negative version of it, you’ll have two solutions. Now, what happens if b squared

minus 4ac is equal to 0? If this is equal to 0– if you

take b squared minus 4, times a, times c, and that’s equal to

0– that tells us that this part of the quadratic formula

is going to be 0, and the square root of 0 is just 0. And then, actually, your only

solution is going to be x is going to be equal to

negative b over 2a. Or another way to think

about it is you only have one solution. So if the discriminant

is equal to 0, you only have one solution. And that solution is actually

going to be the vertex, or the x-coordinate of the vertex,

because you’re going to have a parabola that just touches the

x-axis like that, just touches there, or just touches like

that, just touches at exactly one point, when b squared

minus 4ac is equal to 0. And then the last situation

is if b squared minus 4ac is less than 0. Then over here, you’re going to

get a negative number under the radical. And we saw an example of

that in the last video. If we’re dealing with real

numbers, we can’t take a square root of a negative

number, so this means that we have no real solutions. In the future, you’re going to

see that we will have complex solutions, but if we’re dealing

with real numbers we have no real solution. Because this makes no sense. The square of a negative number,

at least it makes no sense in the real numbers. And then there’s more

you can think about. If we do have a positive

discriminant, if b squared minus 4ac is positive, we can

think about whether the solutions are going to

be rational or not. If this is 2, then we’re going

to have the square root of 2 in our answer, it’s going to

be an irrational answer, or our solutions are going

to be irrational. If b squared minus 4ac is 16,

we know that’s a perfect square, you take the square

root of a perfect square, we’re going to have

a rational answer. Anyway, with all of that talk,

let’s do some examples, because I think that’s

what makes all of these ideas tangible. So let’s say I have the equation

negative x squared plus 3x, minus 6

is equal to 0. And all I’m concerned about is

I just want to know a little bit about what kinds of

solutions this has. I don’t want to necessarily

even solve for x. So if you’re in a situation like

that, I can just look at the discriminant. I can just look at b

squared minus 4ac. So the discriminant here is

what? b squared is 9 minus 4, times a– negative 1– times

c, which is negative 6. So what is this equal to? This negative and that negative

cancel out, but we still have that negative

out there, so it’s 9 minus 4, times 6. This is 9 minus 24, which

is less than 0. So we’re going to have a number

smaller than 0 under the radical. So we have no real solutions. That was this scenario

right here. And so this graph is going to

point downwards, because we have a negative sign there,

so it probably looks like something like that. If that’s the x-axis, the

graph is dipping down. Its vertex is below the x-axis

and it’s downward-opening, so it never intersects

the x-axis. We have no real solutions. Let’s do another one. Let’s say I have– I’ll do this

one in pink– let’s say I have the equation, 5x squared

is equal to 6x. Well, let’s put this in the

form that we’re used to. So let’s subtract 6x from both

sides, and we get 5x squared minus 6x is equal to 0. And let’s calculate

the discriminant. So, we want to get b squared. b squared is negative

6 squared minus 4, times a, times c. Well, where is the c here? There is no c here. There’s a plus 0 that I’m

not writing here. There’s no c. So in this situation,

c is equal to 0. There is no c in

that equation. So times 0. So that all cancels out. Negative 6 squared

is positive 36. The discriminant is positive. You’d have a positive 36 under

the radical right there, so not only is it positive, it’s

also a perfect square. So this tells me that I’m going

to have two solutions. So I’m going to have

two real solutions. And not only are they’re going

to be real, but I also know they’re going to be rational,

because I have the square root of 36. The square root of 36 is

positive or negative 6. I don’t end up with an

irrational number here, so two real solutions that

are also rational. This is this scenario

right there. And you could also have

irrational in this scenario, so it’s this [? here ?] plus the irrational. Let’s do a couple more, just

to get really warmed. Let’s say I have 41x squared

minus 31x, minus 52 is equal to 0. Once again, I just want to

think about what type of solution I might be

dealing with. So b squared minus 4ac. b squared. Negative the 31 squared minus

4, times a, times 41, times c– times negative 52. So what do I have here? This is going to be a

positive 31 squared. The negative times

the negative, these are both positive. So I’m going to have

a positive, right? This is the same thing as 31

squared, plus– this is a positive number right here, I

mean, we could calculate it, but it’s 4 times 41, times 52. All I care about is my

discriminant is positive. It is greater than 0,

so that means I have two real solutions. And we could think about whether

this is some type of perfect square. I don’t know. I’m not going to do it here. That would take a little

bit of computation. So we know they’re real, we

don’t know if they’re rational or irrational solutions. Let’s do one more of these. Let’s say I have x squared

minus 8x, plus 16 is equal to 0. Once again, let’s look

at the discriminant. b squared, that’s negative 8

squared minus 4, times a, which is 1, times

c, which is 16. This is equal to 64 minus

64, which is equal to 0. So we only have one solution,

and by definition it’s going to be rational. I mean, you could actually

look at it right here. It’s x minus 4, times x

minus 4 is equal to 0. The one solution is x

equal to positive 4. And when I say by definition of

the quadratic formula, you look there, if this is a 0,

all you’re left with is negative b over 2a, which is

definitely going to be rational, assuming you have a,

b, and c are, of course, rational numbers. Anyway, hopefully you

found that useful. It’s a quick way. You don’t have to go all the way

to solving the solution, you just want to have to say

what types of solutions or how many solutions, how many real

solutions, or inspect whether they’re real or rational. The discriminant can be kind

of a useful shortcut. And I also think it makes you

kind of appreciate the parts of the quadratic formula

a little bit better.

thankss this video helped alot

man it helped so much that i feel like not going to school anymore

@ABCba5tard i no rite?

wooooohooo imaginary numbers!!!! u should add imaginary numbers to your linear algebra playlist. 'cause it would help me with my college course! especially rref'ing a system composed of imaginary numbers

Well done! What S/W and H/W set-up do you use as a "black board"? It's very easy on the eyes!

I really like your videos Sal, they are very useful for me. Quick comment:

Instead of saying discriminant=zero means one solution, I know it would have saved me many hours of research if someone told me that we still have two solutions (like always with quadratics) but in this case the solution repeats two times. Eg: x^2-4*x+4, even though it's discriminant is zero, it still has two solutions, which are x=2 and x=2.

Again great videos and I hope you do more calculus and physics videos!

When the discriminant is zero there is technically one solution but isn't it a double root?

thanks bro

Aced my final. thanks breh! u saved me from military school.

@loversjustkeeploven really? thats pretty awesome

@paulceltics yea seriously. i live alone but my parents still support me. and they would have made me enlist of i didnt end the my junior year with C's.

@Rowan9789 if 5x^2-6x=0 the constant is zero. you could feasibly write the same equation as 5x^2-6x+0=0. either way, the y-intercept would be 0, as would the x-intercept.

Well… the consistancy in your video's is through the roof. i've just started A levels after getting a good grade in Math in Gcse … there is quite some confusion among the topics i'm studying in this new curriculum… some of the topics i've had trouble in I've searched for help, YOU have covered them all. Thank you.

IF ONLY MY MATH TEACHER TAUGHT LIKE THIS -____-

You are way better than my math teacher, and I wish you were my science teacher because he is just so… argh. =__=

@xcelpast Please don't say bro. :l It's just, just no..

Thumbs up if you have an A in Algebra I! π

why I cant find this video in the website??

My teacher: ***stacks piles of papers on my desk*** DO IT!

You: OK… Let's go over this slowly

YOUR THE BEST!! I ACTUALLY UNDERSTAND WHAT I'M DOING IN MATH CLASS NOW!!! π

how does two solution look like in graph ??

It touches the x-axis in two places, so the parabola goes down touches one point on the x-axis and then when it comes up again it touches a different point on the x axis.

The discriminant of the 3rd equation is 9489.000000000002

it's what BRO? huh whats wrong with saying BRO

I'm quite surprised that you replied to me after 11 months. Smh..

dude, your point is?

I don't have a point. My comment had nothing to with you. Now good day, sir.

aiight bro. cya

Thank you dude! I fell asleep in class. Watching your videos are going to help me get through Algebra 1.

I learned in 2 minutes of this video what took me 1 hour in class to get confused over

how did u get 4?!!!!

oh never mind!!!!!! π

You save my math life thank you so much

How will I know if it is rational or irrational ?? And how will I know if it is complex ??

I gotta give it to you, I learned more from you than my teacher

i dont think i would pass tests without you. cheers mate

Thank you!

oh my god, thank you so much for this because math has not been life.

Very helpful. Thanks!

omg what would happen if khan academy dies, then i would fail every maths test LOVE YOU KHAN ACADEMY KEEP IT UP

SO HELPFUL! Thank you! <3

what real life application does using the discriminant have?

Why am I learning this in algebra 1 if this is algebra 2

Great video straight to the point and clear, great job

…

THANK YOU THIS WAS VERY HELPFUL.. NOW IΒ FEEL LIE IM GOING TO PASS MY ALGEBRA 1 EXAM TOMORROW JUNE9, 2016…. SUMMER IS ALMOST HERE… CAN I GET A THUMBS UP!!!

What about for all real numbers?

the third example is not a perfect square, it is 7567 and the square root of that is 86.98850499…

wait,wait

so where do you get on 5:24

(the orange color)

there is discriminant

so where did you get the 9-4? ( Discriminant 9-4(1)(6) )

thanks it's help me

FUCKINF idiot

I learned more from you than my Teachers way less confusing

THANK YOU SOOOOO MUCH (:

So I put the "i" at the end of the solution in Aleks when the answer is negative?

amazing

khan academy just makes my life easier. My teacher makes everything all complicated but Sal just makes it simple and easy to understand. THANK YOU KHAN ACADEMY

Hey why you didnt put 4(-1)(-6) ? At 6:00

Can anyone give me an example of discriminant

hold up ,wait! u lost me….arnt you suppose to use the quadratic formula??

everyone forgets that Kahn is the first (unintentional) asmrtist

I understand the lesson now. THX But at first, I was like what?. I didn't get anything My teacher usually gets mad when I ask too many questions .

thank you for the arabic translation

If there is cubic equation?

With the internet, who needs school?

JOE LIKES GUYS

this video sucks do something else you suck get off or play fortnite battle royale boiiiiiii

hi brennan

my teacher just takes a sick day and tells us to watch these videos.

Is B always negative?

Does that mean If the Discriminant is 0 there is only 1 solution ?

HELP PLEASE

Every time I finish a video is like "this is so easy how did i not understand it before"

Good video

Whos here from SAS ?! π³

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μ λ°°μ°κ³ κ°λλ€

THANK YOU THIS IS SO GOOD

but I've a question how does it matter if it is a rational solution or an irrational solution

omyghad, thankyouuu!!! β€οΈ

Tangina exam na naman hays.

The amount of OOOOHHHH!!!! moments I had after watching this…….

This is very good i loved it..but i have a question..how did we explore the formula of discriminant ?

What if you have to solve for no real solutions with [i]

Anyone else's teacher force them to watch this?

Watching this because my teacher got mad and she didn't teach this lesson to us but we're having a quiz about this tomorrow. Thanks a lot.

i can safely say this explained and taught me more than my school.

what if its not equal to 0 and hahs 4 numbers in the equation