okay we just derived the quadratic
formula and so now we're gonna use it
we're gonna use it to solve some
quadratic equations it's a powerful
method because you can solve any
quadratic equation using the formula so
here are a few examples um number one
let's look at let's solve M squared plus
and minus 4 equals to zero and so this
is based on this ax squared plus BX plus
C equals to zero this formula is based
on the the equation being in this set up
so a 0 on the right hand side so in this
case if we modeled it off of their a
would equal to 1 B 1 equal to 1 and C
would equal to negative before so M
instead of X M is gonna equal to
negative B negative 1 plus or minus the
square root of B squared B squared minus
4 times a times C all over 2 times a a
in this case is 1 C in this case is
negative 4 2 times a is 1 now we're just
gonna use order of operations to
evaluate this so M is gonna equal to
negative 1 plus or minus the square root
of negative 1 times think I mean 1 times
1 is positive 1 then you count the
number of negatives we have right here
we have a 1/2 so a negative times a
negative makes this positive 4 times 1
times 4 is 16 16 all over 2 times 1 is 2
so the answer to this negative 1 plus or
minus the square root of 17 over 2 so M
is 1 plus the square root of 16 of 17
over 2 and 1 minus the square root of 17
over 2 all right that's number one
number two is this
number two its 4x squared or x squared
minus 12x plus 9 equals to 0 so let's
solve this one using the quadratic
formula this isn't the right setup so
we'll just say X a is equal to 4
B is negative 12 and C is 9 so X is
negative B negative negative 12 plus or
minus the square root of B squared
negative 12 squared minus the number 4
times a which is 4 times C which is 9
all over 2 times a all right so X is
negative negative 12 which is positive
12 plus or minus the square root of
negative 12 times negative 12 is 144 and
then 4 times 4 times 9 let's see that 16
times 9 9 times 6 is 4 54 9 10 11 12 13
14 it is minus 144 again take a look
here one negative means this whole thing
will remain negative all over 8 so we
get X is equal to 12 plus or minus the
square root of 144 minus 144 is 0 over a
we get 12 plus or minus 0 square root of
0 is just a 0 so we just get 12 over 8
so the whole plus or minus thing went
away and so we get 12 of right you could
divide this by 4 over 4 right so that
would be 3 halves this is interesting
this is a case where we had a quadratic
which gives us two answers but in this
case we only got one and when that
happens this is called a double root or
a double zero so this is what our answer
was here okay we're gonna hang on to
that I mean I'm gonna start making lists
for number one number one our answers
were M is equal to Omega I think I'll
write the answer right here
our solutions were in
is 1 plus or minus the square root of 17
over 2 all right
4 to our answer was just X is equal to
three-halves ok um let's look at number
3 it's x squared plus X plus 1 is 0 ok
so this is ready to go so we say a is 1
B is 1 and C is 1 they're all 1 this
time okay X is equal to negative B plus
or minus the square root of B squared
minus 4 times a times C all over 2 times
a and these are all ones so we get X is
equal to negative 1 plus or minus the
square root 1 squared is 1 minus 4 all
over 2 this is gonna be a negative 1
plus or minus the square root of
negative 3 over 2 negative 1 plus or
minus I square root of 3 over 2 so my
two solutions here are negative 1 plus I
square root of 3 over 2
and they
one minus I square root of three over
two okay so we saw up there number one
we solved a quadratic equation and we
got two answers and they were both real
let me grab my calculator real quick so
I can give us a decimal answer so that
feels better to us let's take this
square root of 17 and see what it turns
out to be 1 plus square root of 17
divided by 2 is 1 1 plus the square root
of 17 divided by 2 is 2 point 5 6 2
point 5 6 about and then was like 1
minus the square root of 17 divided by 2
is negative 1 point 5 negative 1 point 5
6 ok so these were the two answers for
that one
okay then number 2 we solve this and we
found out that this quadratic equation
only had one one answer and then we
solved number three and we saw that this
one had two but they were imaginary so
we got it right here we got two distinct
real roots here we got one real root or
zero or solution here we got two
distinct
imaginary so what's going on here well
this is what's going on here two things
one thing is we can figure out what kind
of answers we're gonna get just by
looking at the discriminant so let me
erase kind of the middle part where you
get all this solution because I wrote
down the gist of it and let's discuss
what is a discriminant a discriminant is
it's this part it is this part of the
quadratic formula so the discriminant is
b squared minus 4ac and so if you figure
out what the discriminant is for each
one of these you can determine what type
of solutions you're going to get so what
b squared minus 4ac is positive you will
always get two distinct real roots or
zeros or solutions if the discriminant
is equal to zero you will get one real
root if the discriminant is negative you
will get two distinct imaginary
solutions or roots okay two distinct
okay so right here let's calculate what
is b squared minus 4ac I've erased it to
Lizzie B was one one minus four times a
times C a was one C was negative four so
we had one plus sixteen we got a
positive 17 if the discriminant is
positive you're gonna get two real
distinct roots that's what happened
there those are real numbers what about
this guy what was this guy's
discriminant it's this part right here
this ended up equaling zero what happens
there
you get one real root
that is exactly what happened there what
is the discriminant for number three
number three it was 1 minus for a
negative 3 the fact that the
discriminant was negative means we would
know going into this we're gonna get two
distinct imaginary roots and that's what
happened there so on the last example
let's just predict what kind of roots
are we're gonna get before we do the
whole thing and then we'll go find them
so let's look at a number I don't know
did I call it number three let's look at
number four this last one by the way you
should commit the quadratic formula to
memory I realize we're in an online
setting but it's not doing yourself any
any service not to just memorize this
because later you'll be in a different
math class and you're gonna have to know
it so X is equal to negative b plus or
minus the square root of b squared minus
4ac all over 2a if I was able to test
you in a proctored setting I would have
had a question on my test that said
write the formula down without using
notes or anything so you should I really
believe you should memorize this let's
see let's do the last one and have a say
where did I leave that where's my little
piece of paper oh here it is by the
quadratic formula to solve it 2x squared
minus 3x minus 5 is 0 okay so let's see
I'm just thinking for a second you know
what this one we could solved by
factoring we could saw by completing a
square and we can solve it by quadratic
formula
let's do quadratic formula but first
let's figure out what kind of answers
we're gonna get first we'll say what's
the discriminant what's the discriminant
it's the e squared minus 4ac what is a
is 2 B is negative 3 C is negative 5 so
let's go through and say what is B
squared B squared would be negative 3
squared minus the number 4 times a times
C what is a it's 2 what is C it's
negative 5 this is a 9 and negative
times a negative makes it positive
this is gonna be 40 and so 49 it's
positive we're gonna get two real roots
that's what we know going into this so
just for practice let's solve this by
let's solve this bye bye bye bye bye
what should we do first let's solve this
by quadratic formula and then let's do
completing the square for practice
okay ready and you could also factor
this it's all gonna give you the same
answers first quadratic formula X is
negative B negative negative 3 plus or
minus the square root of B squared
negative 3 squared minus 4 times a times
C all over 2 times X all right a is 2 C
is negative 5/2 times 2 okay ready
negative negative 3 is a 3 plus or minus
the square root of 9 plus 20 times 2 is
40 all over 4 this is gonna be 3 plus or
minus the square root of 49 over 4 this
is 3 plus or minus 7 over 4 here's my
two answers 3 plus 7 over 4 3 minus 7
over 4 this is a 10 over 4 or a 5 halves
this is a negative 4 over 4 or an
my two answers are five halves a
negative one they're both real and we
knew that going into it because we had
looked at the discriminant so let's just
record that you know what I think I
would rather do factoring on this one
let's do factoring just to just for
practice instead of completing the
square or we could do both it doesn't
matter well do we'll do both and if you
don't want to watch it you just don't
have to let's see how about if we did by
factoring this is a good one to factor
2x squared because you'll have to be
able to do whatever method you're asked
to do I'm gonna use AC method a times C
is negative 10 I'm gonna come up with
two factors that multiply to be negative
10 but add to be negative 3 so 5 times 2
is 10 and if I make the 5 negative this
Plus this is negative 3 this times this
is negative 10 that's gonna work so I
need to rewrite my middle term here into
what I chose that to be 2x squared I'm
gonna say minus 5x plus 2x minus 5 is 0
or if you wanted to switch these and say
plus 2x minus 5x it wouldn't matter what
order you write them down it's either
way it's gonna factor correctly so if we
wrote it down that way that's fine what
we could do here is its group the first
two and group the second two make sure
you include that negative there factor
out the 2x leaving you with that X plus
a 1 factor out a negative 5 leaving you
with the same factor X plus 1 factor out
X plus 1
leaving you with 2x minus I said each
factor equal to 0 and saw we'll get X is
negative 1 add 5/2
we get five halves of negative one okay
lastly real quick by completing the
square 2x squared minus 3x minus 5 is 0
move C to the right hand side if a is
not 1 divided by a divide both sides of
this by 2 because a is 2 so we get x
squared minus 3 halves X is equal to 5
over 2 all right now we need to add need
to add B over 2 quantity square to this
after you've done that division so we're
going to add negative 3 over 2 divided
by 2 you could think that it's a tuber 1
so negative 3 over 2 times the 1/2
negative 3 over 4 is what B over 2
equals 2 so we're going to add this
squared to both sides so we're gonna say
plus negative 3 over 4 squared and if I
go ahead and square that's gonna give me
9 over 16 this will always factor into X
plus whatever I see in there quantity
squared over here I'm gonna add these I
need to multiply this times 8 times 8 so
that's gonna be a 40 over 16 plus 9 over
16 so I get X minus 3 over 4 squared is
49 over 16 take the square root of both
sides okay the square root of the left
is equal to plus or minus the square
root of right so we'll get X minus 3/4
is equal to plus or minus the square
root of 49 is 7 the square root of 16 is
4 all right I'm running out of room here
X it's 3/4 plus or minus 7 over 4
3 plus 7 over 4 is 10 over 4 which is my
5 halves 3 over 4 minus 7 over 4 is a
negative 4 over 4 this is my negative 1
so you get the same answer whichever
method that you choose to do it although
I will test you on knowing all of them
methods
