## Discriminant of quadratic equations | Polynomial and rational functions | Algebra II | Khan Academy

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
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
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.

## 84 thoughts on “Discriminant of quadratic equations | Polynomial and rational functions | Algebra II | Khan Academy”

1. gucci jessi says:

thankss this video helped alot

2. Allen Deng says:

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

3. Allen Deng says:

@ABCba5tard i no rite?

4. Dusevic says:

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

5. elbay2 says:

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

6. noblessus says:

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!

7. hockeyr5 says:

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

8. xcelpast says:

thanks bro

9. Mace Rapp says:

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

10. paulceltics says:

@loversjustkeeploven really? thats pretty awesome

11. Mace Rapp says:

@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.

12. twominutepenalty says:

@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.

13. Justmake says:

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.

14. IcedLipGloss says:

IF ONLY MY MATH TEACHER TAUGHT LIKE THIS -____-

15. Jihoon Choe Music says:

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

16. Jihoon Choe Music says:

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

17. Jihoon Choe Music says:

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

18. noobkosh says:

why I cant find this video in the website??

19. todayjanuary2012 says:

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

20. rascaldevilz says:

how does two solution look like in graph ??

21. Crazer says:

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.

22. Joe Jones says:

The discriminant of the 3rd equation is 9489.000000000002

23. J says:

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

24. Jihoon Choe Music says:

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

25. J says:

26. Jihoon Choe Music says:

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

27. J says:

aiight bro. cya

28. Ace Clarkson says:

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

29. crazyloco97 says:

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

30. alex accosta says:

how did u get 4?!!!!

31. alex accosta says:

oh never mind!!!!!! π

32. SainT Wreckless R19 Hacks Tips and Hints says:

You save my math life thank you so much

33. Jacob Rodriguez says:

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

34. Nate Kosty says:

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

35. Ben Walker says:

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

36. Ryan Hong says:

Thank you!

37. Asha A. says:

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

38. A W says:

39. Justin Zhao says:

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

40. Jamie G says:

41. oJose says:

what real life application does using the discriminant have?

42. lookinglasstie says:

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

43. thePostman Delivers. says:

Great video straight to the point and clear, great job

44. Alisia Pride says:

45. Alisia Pride says:

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!!!

46. Jyavant says:

What about for all real numbers?

47. Hans anderson says:

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

48. Gabc Sensei says:

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) )

49. Sonoko says:

thanks it's help me

FUCKINF idiot

51. Flameplays says:

I learned more from you than my Teachers way less confusing

52. Ellle says:

THANK YOU SOOOOO MUCH (:

53. I Donut Care says:

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

54. shobha rani p l says:

amazing

55. aucool786 says:

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

56. V Tech says:

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

57. bisma irfan says:

Can anyone give me an example of discriminant

58. Fritzlanda Andre says:

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

59. Ghast505 says:

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 .

61. Ruba Ahmed says:

thank you for the arabic translation

If there is cubic equation?

63. Itz FlameZ says:

With the internet, who needs school?

64. FreedumFries says:

JOE LIKES GUYS

65. Brennen Felten says:

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

66. Michael Ambroz says:

hi brennan

67. Alex Fischer says:

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

Is B always negative?

69. Random Viewer says:

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

70. Gracie-Animations says:

71. Futurine says:

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

72. Siddharth Buddha says:

Good video

73. Brandt Li says:

Whos here from SAS ?! π³

74. Eduu Ysmm says:

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μ λ°°μ°κ³  κ°λλ€

75. Nobody Iknow says:

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

76. Monica Aira Monserate says:

77. the lazyone says:

Tangina exam na naman hays.

78. aRPGenius says:

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

79. Yamen Nazer says:

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

80. Valerie Crespo says:

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

81. faraz lotfi says:

Anyone else's teacher force them to watch this?

82. Stacey says:

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.

83. sen says:

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

84. Pancake Kingdom says:

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