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The Quadratic Formula & The Discriminant
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Feb 19, 2025
Section 4.6 In our previous sessions, we learned how to use the square root property and the completing the square method to solve second degree equations. In this section, we'll be introduced to the quadratic formula and its use in solving quadratic equations. We'll also talk about the role of the discriminant in determining the number and types of solutions of quadratic equations. Chapters: 00:00 Introduction 03:16 The quadratic formula 04:04 Quadratic formula in action 14:40 Quadratic equation with one rational root 17:13 Quadratic equation with complex roots 20:06 The discriminant
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0:00
so we learned about three techniques to
0:04
solve quadratic equations right the
0:08
first one was we did we did the square
0:10
root property that was last two weeks
0:13
ago and then last week we worked on
0:15
completing the square which I think you
0:18
guys fairly did well on that so I still
0:19
I want to give you quiz on completing
0:21
the square because that's not I and
0:23
complex numbers right just thank God I'm
0:25
not doing the test because that would be
0:27
more question question are better less
0:29
Point better
0:30
now we're going to learn a new technique
0:32
it's called the quadratic formula now
0:35
even if I didn't teach you the square
0:36
root property this one I have to teach
0:39
you because it's going to follow you in
0:40
Algebra 2 no pre-calculus it's going to
0:43
follow you in calculus it's going to
0:44
follow you in chem physics it's going to
0:47
follow you for a long time so I know you
0:51
guys know heard about quadratic formula
0:53
right b square minus all that stuff you
0:55
heard about it so that's going to follow
0:56
you for a long time so I can't not not
1:00
teach it right so first thing we need to
1:03
understand is this what is the quadratic
1:05
equation we've already talked about
1:07
quadratic equation a quadratic equation
1:09
is any equation that is
1:12
written as
1:14
ax² plus b x + C = 0 this is called a
1:18
quadratic equation you can also call it
1:21
a second degree equation second degree
1:24
because this has like the highest degree
1:27
here is two so quadratic equation second
1:29
degree equation they are sys I'm saying
1:32
the same thing all right now example of
1:36
a quadratic equation 2x2 + 8 x + 1 = 0
1:40
that's a quadratic
1:41
equation X2 + 2x + 1 = 0 that's a
1:44
quadratic equation now after this
1:47
chapter if I give you uh like a question
1:49
like this solve this if I don't specify
1:51
the technique to use you can use the
1:53
quadratic equation you can also use a
1:55
square root property so how could you
1:57
solve this here yeah
2:01
um if you let's say you never heard
2:03
about the quadratic equation how would
2:05
you solve X2 + 2x + 1 excuse me subtract
2:08
one yeah but what would be a faster way
2:11
to do it you do the X plus
2:14
one wait X+
2:17
1al zero right like this and then you
2:20
can just solve x be equal to1 right so
2:23
you can use that cuz you can Factor what
2:26
what two numbers multiply up to one that
2:28
add up to two wait Ming the first one or
2:30
the second one no the second one oh okay
2:33
right second one right now now that we
2:37
we we could do that right it's easier to
2:38
just Factor it you can solve this by
2:41
factoring right but let's say now we are
2:43
trying to solve the second one that's of
2:46
top how would you solve it now it's not
2:48
as objects right so we have to use what
2:50
we call the quadratic formula I learned
2:54
this and I know by heart now right
2:56
because I've seen it for years I
2:58
remember when I was in M College as a
3:00
student I was introduced to the
3:01
quadratic formula actually back in too
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when I was in uh 10th grade I was
3:06
introduced to the quadratic formula you
3:07
stay with me all these years because you
3:10
have to use it constantly right so the
3:13
quadratic formula the solutions of a
3:16
quadratic equation can be found using
3:19
the quadratic formula so if you have an
3:22
equation in the form ax² + BX + c = z to
3:26
solve it you can just apply this formula
3:29
I know you guys no formula so you can
3:31
use this formula to to solve it all
3:33
right so X = B plus or minus b^2 - 4 a 2
3:39
a now it's easy to to say but how do you
3:44
apply the quadratic formula I'm glad you
3:46
asked right so we're going to solve it
3:48
let's say we can try to solve this
3:50
problem here I have
3:52
2x2
3:54
right + 8
3:57
x + 1 = 0 and I want to solve this
4:00
equation using the quadratic formula
4:02
right my first step is to identify A and
4:06
B and C right a is equal to what four
4:11
two huh two two B is eight and then C is
4:15
one one now we have to plug all these
4:17
values in here now I'm sure your ti3 is
4:21
sophisticated and I'm sure that is a key
4:24
a way to solve this if you can use the
4:26
T3 to put it in your calculator go for
4:28
it right go for it if you can I'm not
4:31
going to be mad because I know you use
4:33
the quadratic formula
4:35
so somebody can even look that up online
4:38
at some point and then come up with like
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this is how use the quadratic formula to
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solve equation right so now I'm just
4:44
going to replace it NE B right B will be
4:48
what 8 correct cuz B is 8 plus or minus
4:54
b² that is H squ now notice what I do
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here I'm going to put stuff in
4:58
parenthesis to avoid problems so I'm
5:01
going to put B b² parentheses right - 4
5:05
* a a is what 2 and C is 1 so I'm going
5:10
to put all this in parentheses over 2 a
5:12
which is 2 * 2 right now the rest is
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knowing how to compute the stuff that's
5:20
under the radical because we have to
5:22
learn to simplify so x = -8 Plus orus 8
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s is 64
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and then 4 * 2 is 8 8 and 1 is 8 so this
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is 64 - 8 right over 4 now I'm going to
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simplify
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that so I get x
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= 8+ orus 64 - 8 that is 50
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what 50 oh my God 56 thank
5:52
you right over four right and now look
5:57
I'm going to split my answers now
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I'm going to have x
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= -8 + 56 < TK over 4 or X also equal 8
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- 56 over 4 right now yeah now you're
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going to be pra I know you're not going
6:18
to like it but we have to simplify this
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14.7 we can't have uh decimals you know
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when it com to this 15 we have to keep
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no 56 is broken down into what
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four per square four and what 4 and 14
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right so it's going to be x = 8 + 4 * 14
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/ 4 and x = 8 - 4 * 14 over 4 now I know
6:49
what of 4 is what is it two so that's is
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x = -8 + 2 < TK 14 / 4 and X =
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8 - 2K 14 over 4 now be very careful
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here right we always have to simplify
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our answers now if I see I have 8 2 and
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four 8 2 and four right so now I can
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simplify this now a lot of people last
7:16
year when we were simplifying radicals
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they also try to simplify what's inside
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the radical you cannot do that because
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you can't do that that's just cannot be
7:25
done all right so what goes into eight
7:29
and two and four two two so if you
7:32
simplify that you're going to get here
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-4 plusun 14 / 2 and here you're going
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to get x = 8
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-4 by 14 / 2 and then this is when you
7:46
stop all right you stop here so as you
7:50
can see we have to use what we've
7:52
learned in the previous chapter you have
7:54
to be able to simplify square root
7:56
functions or square root numbers if you
7:58
don't know how to do that you're going
7:59
to have a hard time figuring this
8:01
figuring this thing out yes do we have
8:03
to use like numbers like four and 14 cuz
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doesn't like 6 and 9 that's 6 and 9 is
8:10
54 okay right I'm trying to remember I'm
8:14
Al I'm always trying to see it as a
8:15
perfect square inside the number that I
8:17
have so I can simplify it right so
8:20
that's just what it is now do I wish
8:23
there was a different way to solve this
8:24
yes I wish but
8:26
unfortunately guess what woman up man up
8:30
I can't tell you there this is how we
8:32
going to do it right just have to suck
8:34
it up and just get it down I can't do
8:36
nothing else now let's try to solve this
8:39
this equation with this
8:41
right so if I can I erase this oh you
8:44
somebody sit right
8:45
this I no
8:50
no you see right in there all
8:54
right
8:57
no I don't have an extra Mar
9:00
do I look
9:02
like marker
9:04
dealer Mar Tre
9:07
you she have
9:09
x + 5 x + 6 = 0 she was a one B five C
9:17
six all right so I'm going to use this
9:20
right negative B will be negative 1 see
9:23
huh so give me how would you write this
9:27
five mhm plus or minus mhm
9:30
um
9:32
five um what it called
9:36
okay- 4 mhm
9:39
one and six all right and then 2 * thank
9:46
you all right
9:48
yeah
9:52
yeah
9:54
AC oh it's always four yeah it's always
9:57
that's the formula it's always four - b
10:00
2 - 4 a c it's always right never
10:03
question the
10:06
formula what's 5 squ what's 5
10:10
squ all right and Shay Contin what's
10:15
that going to give you here
10:17
um 4 * 1 is wait what 4 * 1
10:26
is4 over two right now look this is
10:29
actually perfect here now I have this
10:31
that's just it right yeah Ro TK of 1
10:33
over two now what root of one Z one who
10:37
say
10:41
zero now we
10:43
have plus orus 1/ two and now we can
10:46
split the answers all right so I always
10:49
like when the number inside of here is a
10:51
perfect square so that way I don't have
10:53
to Z like trying to simplify it we have
10:55
to split the answers or can we just
10:57
write plus or minus well when you have
10:58
this you can go ahead and then finish it
11:00
because you know what 5 + 1 is is4 over2
11:03
and then -6 over2 so you don't want to
11:05
stop here you want to go and just get me
11:08
the5 + 1 2 and - 1 2 right how would you
11:14
how would you how would you tell the
11:16
difference between the right X and the
11:18
wrong X is the x is the X6 uh both will
11:22
work if you subtitute them in the
11:23
equation you have you have two answers
11:25
when dealing with squares the number can
11:27
be netive or POS
11:29
because it's
11:37
it's I had the answer I forgot to write
11:40
it's 2 which is2 and then
11:43
6 yeah yeah I
11:51
know you know what it is you don't stop
11:54
you know what it is you can find right
11:56
so all you have to just go an extra step
11:59
now does the calculator have a way of
12:01
doing this I think there is no I got I
12:03
got an error well the problem is I think
12:06
the reason why you got an error is
12:07
probably that way you put it in but I'm
12:08
sure with how sophisticated the T3 is
12:12
there the way to find it now if you can
12:14
set up the problem and put it in
12:15
correctly then you be able to find the
12:19
answer there's got to be something if
12:21
you can find it out that' be great right
12:23
you can't do plus or minus it would give
12:24
you
12:25
result put it in two separate ways all
12:28
right
12:29
but if you put a fraction in work I want
12:32
you all to try this on your own right
12:35
try to solve this on your own and tell
12:36
me what you
12:46
got I want you to try this on your own
12:48
right so say you
12:51
have x²
12:54
right + 8 x + 16 is equal to 0 I want
12:58
you to use a quadratic formula and tell
13:00
me what you've gotten what's your answer
13:02
we using quadratic equation yeah
13:04
quadratic equation is here quadratic
13:06
formula this is the quadratic formula
13:08
this is called a quadratic equation
13:10
that's the equation or that's the
13:11
formula right so plug it into the
13:13
formula and see what's your
13:17
answer so we have X2 + 8 x + 16 is equal
13:21
0 and we want to use a qu you figure
13:23
this out I'm not sure
13:27
Quadra yeah it works yeah what' you get
13:30
wait did you when when putting it in
13:32
directly you have to choose plus or
13:34
minus but it will give you the answer if
13:35
you put it in correctly the issue is you
13:37
have to put it so5 over 2 plus Square <
13:42
TK of parentheses 5^ s minus 4
13:46
parentheses 1 parentheses 6 and
13:47
parentheses on top of the parentheses
13:49
after the six to make it all
13:52
part fast and then put it over to I
13:55
think you know what I'll do is I'll try
13:57
to see if I can find coding it's a
13:59
coding thing you got to yeah I I trying
14:02
to go look it up online so I can make
14:05
your life a little easy so you don't
14:06
have to like strug to do it manually
14:09
right this work batteries are dead oh
14:12
wait what I got you new
14:16
64 okay what's wrong with that so what
14:18
do we do plus what do you have then what
14:20
do we do plusus Z is what that's cool
14:24
all right so what do we do with what do
14:26
you have tell me write down what you
14:27
have I have4
14:29
over what
14:30
X8 over two which is that's it what
14:35
about minus don't we need plus Z
14:39
question you see I put this on purpose
14:41
what do you have here one rational rout
14:44
all right got a
14:48
negative what it's not my S root a
14:52
negative would it just be then what are
14:56
you doing all right it should be so if
14:58
you do correct you should have this
15:02
right8 plus or- 64 - 64 right did
15:06
anybody get that yeah right so it should
15:09
give you x = 8 right plus orus 0 over 2
15:13
since it's 0 you just disregard it right
15:16
so we only have how many solution here
15:18
one which is4 that's it right whenever
15:22
the number that's under equ zero you
15:24
going to get one this is why we call it
15:27
one
15:29
right you see what happened here right
15:32
64 get rid of the or
15:36
minus oh because there's no square root
15:38
the plus or
15:43
minus we don't need it all right I did
15:47
just solve this in three steps
15:55
no he's got me there that's two steps
15:58
the so now one now let's let's do
16:02
another one here right let's see I I
16:05
have
16:09
[Music]
16:12
this that's
16:14
inconceivable handing out a fre you guys
16:16
ever seen that movie princess The
16:18
Princess Bride yes I love the princess
16:24
[Music]
16:29
now tell me something here right I have
16:31
x 6
16:34
= thank you very much you call it right
16:37
away right this is not in the right what
16:39
in the right format I have to put it
16:41
right so that's why he said I have to
16:43
add what 10 right so now I have X2 - 6 x
16:46
+ 10 is = you have to make sure you put
16:50
it back in the right format before you
16:53
solve it cuz if you don't do that people
16:55
are going to be like oh c is equal to
16:57
-10 you don't want to do that you want
16:58
to make sure it's in the right quadratic
17:02
equ written as a quadratic equation yeah
17:05
the is already negative we put into the
17:07
formula become positive positive that's
17:09
correct now if you go here right
17:11
negative B so what would be negative B
17:13
it would be negative negative what 6
17:16
right plus or minus -6 s notice how I
17:20
put this in parentheses why I put if I
17:24
don't put in parenthesis what's 6 s if I
17:26
don't put in parenthesis it give you
17:27
what3
17:30
you got to make sure you put in
17:31
parentheses if you don't you're going to
17:32
get it wrong right seems like a Prett
17:34
good test- 4 a a is equal to what here a
17:38
is 1 C is what C is 10 right over 2 * 1
17:44
I got x = 6 plus or minus 6 6 is 36 - 40
17:50
ooh
17:52
problem 6 plus or minus is everybody
17:56
here
17:59
all
18:05
right 2 i m yeah oh I went further than
18:10
that I got three plus orus I yeah you
18:13
simplifi div yeah good right if you get
18:17
to four we've already talked about
18:21
radical that have netive under we have
18:22
to put out water I right so this will
18:25
give you what 6 plus or minus 2 i/ 2
18:32
right because of4 is I
18:35
2 we just going to have some I to De
18:40
right what's
18:45
that so we got 6 plusus 2 I 2 already
18:49
got
18:51
that because and then you when you get
18:54
three plus orus I get three plus orus I
18:56
that's correct three plus orus I because
19:00
divisible what can we leave it as 2 I
19:05
you can't but as a grown person you
19:07
should go further because you trying to
19:10
you you need to simplify right in
19:12
mathematics you I'm trying to make this
19:13
work in a calculator
19:16
right IOP here around the 40 and 3 I
19:21
could make it work your culator
19:24
yeah okay is there a plus or minus
19:26
button on no just got do
19:30
either all right so
19:35
now all
19:38
the okay now the next we talk
19:42
about I'm going to talk about this next
19:44
thing and then it's going to be easy
19:46
right we're going to talk about the
19:49
discriminate indiscriminate no the
19:51
discriminate right so there's
19:53
discrimination in that so we're going to
19:56
talk about this guy it's called the disc
19:58
discent right and it's one of the the
20:01
most beautiful thing that you will ever
20:02
see right so why do we need to know the
20:05
discent it helps why is it all caps it
20:09
helps it's not not in figuring
20:13
out figuring out the
20:17
type the
20:20
types and
20:23
number of
20:27
solutions of a quadratic equation of a
20:30
quadratic equation right so you use this
20:33
discriminant to figure out the number
20:36
and the types of solution a quadratic
20:39
equation has right so how you find the
20:41
discri the discriminant is easy we going
20:43
to call in D right d =
20:46
b² - 4 a c so the discriminant is b² - 4
20:51
a c and this is actually very easy if
20:53
you this one you going to have fun if I
20:55
give you a quiz if you see this rub your
20:58
hand because because you know you going
20:59
to have a good grade on that right so
21:02
for example why we use a discriminant we
21:05
use it to figure out the number and
21:06
types of solution an equation has the
21:08
last equation that we worked on was x²
21:11
right + 6x = what -10 so the question
21:15
could be how many solutions is this
21:17
problem going to have right we use the
21:19
discriminant to do this so again we got
21:22
to put this back in the right form which
21:24
is what
21:25
x² + 6 x + 10 =
21:29
now I need to find what is it
21:33
question the X
21:37
come this this is the formula that's the
21:39
formula that's the formula it's called
21:41
the discriminant that's the formula for
21:43
the
21:44
discriminant discriminate formula
21:48
discriminate he's giving us an example a
21:52
formula right so again the definition of
21:54
discrimin it helps in figuring out the
21:56
types and number of solution quadratic
21:59
equation has right so in this equation
22:01
X2 + 6 x + 10 = 0 the question is how
22:04
many solutions and types of solution
22:07
right so you just need to find the
22:08
discriminant what's big
22:10
shap uh 6 so you do 6 s - 4 * a which is
22:16
what 1 C right and then C which is 10 so
22:20
it's pretty much 36 - 40 and that's
22:24
equal
22:24
to4 now since this is negative right
22:29
4 is a negative number so you say4 is
22:32
less than zero because this is less than
22:34
zero this the the the response is this
22:38
equation has two complex solution
22:42
whenever the discriminate is negative
22:44
you have two complex solution let me say
22:46
that again whenever you have a negative
22:49
discriminant how many solutions you have
22:51
two two what comp two complex Solutions
22:53
and then we just figure it out right we
22:55
have 3 plusus I so because this is
22:57
negative
22:59
two complex Solutions and that's it two
23:01
complex
23:03
Solutions
23:05
so if it's if the discriminant is
23:08
negative you have two complex Solutions
23:10
if it's
23:12
-200 huh it's you have two what two real
23:17
solutions right so when
23:21
disc real you have two real solutions
23:24
two complex or one real solution we're
23:26
going to get to that right so is that
23:28
all you can have that's all okay that's
23:30
it that's why I say like this is
23:32
probably the easiest one of all right
23:35
list again be like are we going to do
23:37
this a lot yeah well for a little while
23:40
so if the discriminant is negative you
23:42
have two complex solution right but if
23:44
the discriminant D is positive right
23:48
what I put is
23:49
positive you have what what was the um
23:53
the conclusion when you have a positive
23:54
discriminate two real solutions right
23:58
two real
23:59
solutions right and then the last thing
24:02
is this when the discriminant is equal
24:03
to zero what kind of solution do you
24:07
have one what one real solution when
24:10
discri is zero you have one real
24:14
solution so you only have three option
24:16
three options I mean right if the
24:19
discriminant is negative we have two
24:21
complex Solutions if the discriminant is
24:23
POS we have two real
24:26
solutions and if the disc is equal to Z
24:29
we have one real solution and that's it
24:32
so this chapter should be like I say a
24:37
work that's it
24:40
right so now to do tomorrow
24:47
yes we
24:51
just that's you don't
24:54
find that because you have Sol that's it
24:59
I don't need you to find it if I need
25:01
you to find it I'll tell you to solve
25:03
the equation right but if I say use the
25:06
I'm saying use this to figure out the
25:08
number and type of solution we have
25:11
that's only
25:12
four all right okay can I have a red
25:15
marker no
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