00:00Hi everybody. So in this
00:03lesson we're going to look
00:04at the discriminant, what it
00:05is, and where we use
00:07it. So firstly, in the
00:10formula booklet, section 2 .7,
00:12we are given this, this
00:14is our quadratic equation and
00:16the formula, and this is
00:18the discriminant, B squared minus
00:194C. It is this part
00:22of the quadratic formula. And
00:25it is denoted using
00:28in capital Delta, which is
00:31basically just a triangle like
00:32this, and it is b
00:33squared minus 4ac. Now, what
00:37we use it for is
00:38to decide how many roots
00:41the quadratic function has, or
00:43how many solutions to the
00:44quadratic equation. And the reason
00:47we can use the discriminant,
00:50where it comes from is
00:51this quadratic formula. So imagine,
00:54like this is in the
00:55square root here.
00:56So imagine, let me just
00:57get rid of that. Imagine
01:00B squared minus 4 is
01:01C, or the discriminant is
01:03bigger than 0. But what
01:05happens is, so let's say
01:06we had square root of
01:089, for example, what happens
01:09is you get plus, let's
01:12square root of 9 is
01:133, so you get plus
01:153 and minus 3. So
01:16you get 2 distinct, and
01:18that's the word I'm going
01:19to use down here, distinct,
01:20you get 2 distinct real
01:22roots. One of them is,
01:24plus three, one of them
01:25is minus three, two distinct
01:26meaning completely different roots. If
01:29it equals zero, what happens
01:32is you get this square
01:35root of zero, which is
01:36plus zero minus zero, which
01:39is actually the same thing.
01:40So you effectively only get
01:41one root or one solution.
01:44Now we call it, instead
01:46of saying one root, we
01:48say two equal roots, which
01:50is a bit silly, but
01:51that's
01:52What we say, are we
01:53say one repeated root? I'll
01:55explain more what that means
01:56graphically down here, but let's
01:58just go through this. So
02:00if this equals zero, we
02:02have one root are two
02:03equals. And then if this
02:05is less than zero, what
02:07happens is, let's say we
02:08have the square root of
02:09negative nine, well, the square
02:11root of negative nine, we
02:13cannot do. There is no
02:14square root of negative nine.
02:15Unless you start studying complex
02:17numbers in higher level, but
02:20It's not get into that
02:21right now. If the square
02:23root of a negative number,
02:24we cannot do. You cannot
02:25square, if you square a
02:27negative three, you get nine.
02:29You don't get negative nine.
02:30So there's no square root
02:31giving us no roots. Okay,
02:35now what that means graphically,
02:36if the discriminant is greater
02:37than zero, we have two
02:40distinct real roots. Now what
02:42that means is that's your
02:43kind of normal quadratic and
02:44it crosses the x axis
02:46twice. These are your two
02:48distinct roots. If you had
02:50the negative quadratic, it would
02:52go up and down and
02:55it crosses in two places.
02:58I mean, it could be
02:58anywhere. Depends on the quadratic,
03:00obviously. But we are just
03:02focusing here on the number
03:04of roots. So here I
03:05have two distinct roots. If
03:09we have two equal roots
03:10or one repeated root, what
03:11happens is it comes down,
03:14but it, I don't know.
03:16That's pretty drawn. It comes
03:17down, but it turns exactly
03:20on, that's also pretty drawn,
03:24on the x -axis like
03:26this. So it touches the
03:29x -axis in only one
03:30place, or the negative quadratic.
03:36Like so, touches it only
03:39one. So there we have
03:41our one repeated root, or
03:43as I said,
03:44say two equal roots. And
03:46then finally, if there is
03:47no roots, what happens is
03:49it comes down and it
03:50doesn't touch the x -axis
03:51at all. Or the negative
03:54one doesn't touch the x
03:55-axis at all. Like this.
03:58Okay, so let's look at
04:01two examples. The first one
04:03is fairly straightforward and then
04:05the second one is an
04:06exam type question. So first
04:08he says, use the discriminant
04:09to determine the number of
04:10solutions to the equation this
04:12to this
04:12is our equation, how many
04:14solutions does it have? So
04:15it either has 0, 1,
04:17or 2. The way we
04:19find it is using the
04:21discriminant. So I'm going to
04:22say the discriminant, which is
04:24b squared minus 4ac. That's
04:26in the formula booklet, is
04:28equal to, now I always
04:30write at the side a
04:32equals 2, that's my a,
04:34the coefficient of x squared,
04:36b equals negative 3, that's
04:38my b, negative 3, be
04:40careful.
04:40and C is equal to
04:425, it is the constant
04:44that is left over. So
04:47I have B squared minus
04:494 AC, so B is
04:51negative 3 squared minus 4
04:54times A times C, which
04:56is 5, 9 minus 4
05:00times 5 is 20 times
05:012 is 40, 9 minus
05:0340 is negative 31. So
05:07negative 31 is
05:08less than zero, therefore there
05:12is no solutions to the
05:16equation. So the discriminant tells
05:19us how many solutions. So
05:20I know this has no
05:21solutions. And if I was
05:22to draw that, it would
05:23probably, what it would look
05:25something like this, a smiley
05:28face that doesn't touch the
05:29x -axis. I think it's
05:30probably over here somewhere, but
05:32it will not touch the
05:33x -axis. Finally, I'm going
05:36to do a pass
05:36question. This is a little
05:39bit more tricky because they've
05:40included K's in the function.
05:44So here it's as consider
05:44the function this find the
05:47value of K for which
05:48it has two equal roots.
05:50So tells us now there
05:51are two equal roots. Now
05:54when I see these when
05:56I see these words two
05:57equal roots are two repeated
05:59roots or sorry one repeated
06:00roots are no roots that
06:02tells me you need to
06:03get used to seeing when
06:04you
06:04see that you know it's
06:06discriminant. So just for writing
06:09discriminant often you have to
06:10check each mark scheme but
06:11often it says use of
06:14discriminant anywhere is worth a
06:17mark. So if you recognize
06:18that you need to use
06:19the discriminant here you'll get
06:21a mark. So just for
06:22actually if you just drew
06:23this triangle and left the
06:25whole page blank you'd get
06:26a mark. Okay so I
06:29want the discriminant and then
06:31it's a two -equal root
06:32so this
06:33equals zero. In fact, sometimes
06:35I've seen Mark's schemes where
06:36if you just wrote this,
06:38you'd get two marks because
06:39when you've recognized to use
06:41the discriminant and two, you've
06:42recognized that for two equal
06:44roots, it equals zero. So
06:46b squared minus four, a
06:48c equals zero. Now, I
06:51am going to write at
06:54the side, a equals one
06:56minus k. Now, be careful.
06:58A is, it's the coefficient
07:00of
07:01x squared so it's all
07:02of this, it's in a
07:03bracket. B is equal to
07:05one because it's like one
07:07x, that's my B. And
07:09then C is gonna be
07:10K. Sometimes I've seen functions
07:13where it's like plus K
07:15plus 10. And in that
07:17case C is K plus
07:1910, so just be very,
07:20very careful there, right? B
07:21squared is one squared so
07:23that's one minus four times
07:25A is one minus K
07:26and C is K.
07:29And this equals zero. Now
07:31I just have to solve
07:32this. So I'm gonna do
07:33one minus, and I'm just
07:36gonna do this to make
07:37it a little bit easier
07:38to see. I'm gonna join
07:39this. I'm gonna do the
07:404k first. So it's 4k,
07:42one minus k equals zero.
07:44Sometimes students mess up here
07:47when they're multiplying three things.
07:49So try and just break
07:49it down, make it as
07:50easy as possible. So this
07:52is one minus, minus 4k
07:54times one is minus 4k,
07:56and now I've
07:57negative 4k times negative k,
08:00which gives me plus 4k
08:05squared equals zero. Now I
08:07have a new quadratic with
08:08k's. So it's 4k squared
08:11minus 4k, 4k squared minus
08:144k plus 1 equals zero.
08:17I'm going to factorize this.
08:19I'm going to do this
08:20quickly. So it's 2k and
08:242k.
08:25And it's 1 and 1.
08:26Fact is a 4k squared
08:27or 2k and 2k. Fact
08:29is a 1 or 1
08:30and 1. Both signs are
08:31the same. Both signs are
08:33negative if you are not
08:35familiar with what I've done
08:36there. Check out my lesson
08:38on factorizing. OK, let me
08:41just go over here. So
08:442k minus 1 times 2k
08:51minus 1.
08:53equal zero. So the only
08:56thing I have is 2k
08:57minus 1 equals zero. 2k
09:00equals 1 and k equals
09:06a half. So that's the
09:08only solution. Put a box
09:11around it to clear. This
09:13is the only solution that
09:15would work, that would give
09:16me two equal roots and
09:18that's why it's defined the
09:19value, not the values. So
09:21k
09:21a has to equal a
09:23half. Okay, so I went
09:25through that pretty quickly. Obviously,
09:26just watch it again and
09:28slow it down. If you
09:28need to, this particular question,
09:31what is obviously essentially important
09:35that you understand is this.
09:38If it is two distinct
09:40roots, the discriminant is greater
09:42than zero. Two equal roots
09:44are one repeated root, the
09:45discriminant equals zero, and finally
09:47no real roots, the discriminant.
09:49is less than zero.