Hello. In this video. we're going to learn the
quadratic
formula, and learn how to
solve quadratic;
in particular,
our goal
is to be able to solve
quadratic equations
that look like this,
a quadratic expression
set equal to zero;
and you might think, "That's really
limited; what if you want to
set a quadratic equal to something
other than zero?"
Don't worry, we'll show an example
where we do that.
But let's just dive in
with the method; the quadratic
formula says that if you have this a, this b, this c,
set equal to zero,
then x is (negative b)
plus or minus the square root
of (b squared
minus 4 times a times c)
divided by (2 times a)
and this is the quadratic formula.
It's a messy thing.
Let's make a couple of observations;
first of all, in general, we're going to
get
two solutions when you use this
formula; addition gives us 
a solution, and then
subtraction gives us a solution.
However we do have this square,
root there's no worries in
that b^2 - 4ac
ought to be positive, I mean no
inherent
reason, so you might sometimes
get complex solutions
using the quadratic formula.
Let's start simple,
and by "simple"
I mean that all of our coefficients
will just be in nice
integers and
we don't have any complex
numbers or anything like
that so let's say we want
to solve 2x
squared plus five x
plus two equals
zero
well there's not a lot to really say
about this in a sense it's extremely
thug and fey
um x equals
b is five so negative
five plus or minus
the square root
of five squared
b squared minus
four times
a a is two
here
times c c is also
two
all divided by two
time is a a
is a two so we just
take this a this b and this c
and plug them where appropriate
into this formula
and negative five plus or minus
okay five squared is
25 minus 4 times
2 is 8 times 2
is 16
all divided by
four so
simplifying that square
root further
i selected these coefficients
complete the at random
ah by coefficients i mean to say this b
and this c but i seem to have lucked
into quite
a nice situation
the square root of nine isn't
some kind of ugly decimal
it's just three usually
the quadratic formula is going to give
us decimals
so you see how we get two solutions from
this negative five minus
three divided by
four
negative five thus
three divided by
four and these
simplify as what the negative
5 minus 3 is negative 8
divided by 4 is negative
two negative five
plus three is negative
two
so i'm genuinely kind of
uh stunned
that we got such nice solutions from a
randomly selected
quadratic um negative two squared is
four times two is eight
minus ten is negative two
plus two is zero
negative two really is a solution
and negative one half as well
let's do an obvious problem
let me tear this out
so that we can refer back to the
quadratic
formula when we need
to
let's do the problem
that we failed to do
earlier when we were looking
at factoring
this is a this is b
this is a c
and although our um
our coefficients are rather
uglier than they were
in the previous example
um it's still plug and
play
with this formula
maybe let me try to arrange
things so that we can
see both the formula
and the problem
x equals negative
b plus or minus
the square root
of a b squared
minus 4
times a
times a c
divided by two
times a
and i know i'm going a little quickly
but as i said
we're just taking
this a b and c
and everywhere they appear in this
formula we plug them in
so like why do we have a 10 squared here
well what's from the b squared
b equals a 10
and now we're going to take
this and plug it
into our calculator
so when things go wrong
they usually go wrong
here just because this is such an ugly
looking
thing there are several places you could
make a mistake
um the first thing to recognize
is that your calctivator is going
to follow the order of operations
if you type negative 10 plus
this square root
divided by this denominator
you have made an error already
the cultivator is going to do the
division
first and then it will do
this edition what we
want is for the addition to be done
first everything that's happening in the
numerator
should happen then
we divide
so grouping comes before
division if we put parentheses around
the numerator we should be fine we
don't have up thus one minus
key we're just going to have to do these
one at a time
so for a ti-84
plus silver and
some other additions i suppose
um once you put the square root
once you type the square root in
you'll be taken under the square
root in some earlier calculators like a
ti-84
you won't be you'll press the square
root and your calculator will just
create that symbol
and then you'll have to put parentheses
in
to know um so that your calctivator
knows what it's taking the square root
of
but let's see now 10
squared minus
4 times negative
4.9
times 7.8
get out of this square root
what's this parentheses remember
we've got the entire um numerator in
parenthesis
so we're closing those parentheses
here divided by
once again you have to have parentheses
your cultivator will do order of
operations and you
don't want it to do with division
before multiplication
so you use grouping to control
that
we get negative .583
seconds i'll have
more to say about
this sorry please go away
thank you but um
first let's get the other solution
we're going to get the other solution
with the less
hassle then we got the first
solution thanks to the
entry command we press the blue
second button
then we press enter and that
brings this back up and now we can
scroll around and
edit this
and we'll turn this edition
into subtraction
to get the other solution
now what often happens when you have
word problems involving
quadratics
is that you'll get multiple solutions
to solutions but only
one of them will make
sense
so um
when will the object hit the ground
you're
throwing the object off a platform
it hits the ground here
over here
in the second quadrant
the parabola is equal to
zero but that
negative solution doesn't
make sense i mean time
has to be positive
we are as is so often the case
stuck in the first quadrant
so we'll cross out that solution
it doesn't make any sense the object
can't hit the ground
after a negative number of
seconds
and we'll be left with only one
solution that works
very well um i think for time reasons
i'll end this video here and then do a
second uh set of examples in another
video rather than try to
cram them all into this
one
