- [Instructor] Hello, so
this video is gonna cover
solving quadratic equations by
using the Quadratic Formula.
So then, the first thing
that we need to know is,
first off, what is a quadratic equation?
Alright, so over here on
the left side of the screen,
I have a quadratic equation.
And essentially a quadratic
equation is an equation
that has a square term in
it, so this ax squared here,
is one of the main things that
you should be focusing on,
and as soon as you see that
little exponent of a two,
you know, hopefully bells are ringing off,
hey, this is one of those
quadratic equations.
Now there are a few
different ways that we can
solve a quadratic equation.
The most preferred method,
in my opinion, is by factoring.
And if we can factor it, we
should always do that first.
But not every equation can be factored,
so if it can't be factored,
our other options are to use
the lovely Quadratic Formula,
or by completing the square.
So this video is gonna focus
on using the Quadratic Formula.
So if we look at that
Quadratic Formula down there,
right, what a lovely formula,
x equals negative b plus
or minus the square root of
b squared minus 4ac, all over 2a.
Very, very famous formula here.
If your teacher does not
allow you to have a sheet
or anything like that, and you
have to memorize this thing,
hey there are all sorts
of songs and chants
and all sorts of stuff
online that you can find
that might help you with that.
But if we're looking at this formula,
I call this a plug and chug problem.
Alright, a plug and chug
problem is when we're given
values to plug in, and
we're given a formula,
and we're plugging things into it,
so we need to simplify it
down and get an answer.
So I call the Quadratic Formula
just a plug and chug formula.
And looking at the formula,
we have Bs in the formula, we have As,
and then we also have a C.
So the main thing is first, well hey,
where does that A, B, and C come from?
So going back up here to
the quadratic equation,
notice there is an a,
b, c in that formula,
and those are important.
Now a- a is always in front
of the x squared term.
So that's always going to be your a.
So if I'm looking at the example
that I have on the screen
right here, right, 2x squared plus 3x
minus five equals zero.
Well my a is gonna be the two,
because the two is what's in
front of the x squared term.
And that's gonna be true every time.
Look at the x squared term,
whatever's in front of it,
hey, that's your a.
B then, is always gonna
be in front of your x
to the first power term.
So taking a look at my example here,
here's my x to the first power,
and in front of that I have a three.
So b is going to be a three.
And then c is gonna be your constant term,
so looking over here I have a five,
do note that that is a negative five,
signs do carry, right?
The two and the three
over here were positive,
no need to write a plus
sign in front of it.
But because that five has
a minus in front of it,
we do need to tack on that
negative along with that five.
So for this particular example,
a is two, b is three, c is negative five.
Do be aware when it does
come to certain examples
that you're given, you
could possibly be missing
one of those terms, maybe
the middle term is not there.
And if you do have a situation like that,
then that just means well,
you'd have a zero for that.
So let's pretend this was just 2x squared
minus five equals zero,
well my b would be a zero
then, for that example.
So just be aware that that could happen.
So moving on then, after we
figure out what a, b, and c is,
we need to plug into the formula.
So we are solving for x,
so x is going to equal,
starting with our numerator,
a negative b.
So the negative is part of the formula,
and b is a three in this example.
Plus or minus the square root of
b squared, so b is three,
so that's a three squared,
minus four, times a, which
is a two in our problem,
times c, which is a negative five.
So that's our numerator.
All over 2a, so two times a,
which for us here is a two times a two.
Now I always recommend starting with
what's underneath a radical here,
so this is really just one big
Order of Operations problem,
so hopefully we do remember our
Please Excuse My Dear Aunt Sally,
which just tells us the order in which
to simplify down an expression.
So we almost wanna treat that radical
as if it were parenthesis,
and we wanna get rid of
everything that we have in there.
So I'm just gonna move over the x equals.
I'm gonna drop down that
negative three for now,
and the plus or minus sign,
we're not messing with that quite yet.
We want to simply what's
underneath that radical.
So underneath that radical, right,
following Order of Operations
we would have to take care of
exponents first, so we would
take care of the three squared,
which gives us a nine.
Then moving on to the rest
of what we have there,
so the rest of what we
have there is actually,
we can look at this as a
negative four times two
times negative five.
Alright, so this is all
being multiplied together.
So, negative four times
two is a negative eight,
and then negative eight
times negative five is a positive 40.
So we're gonna say plus 40 right there.
You can also look,
you could have also dropped
down the minus sign,
and did four times two is eight,
eight times negative five is negative 40,
and then you would've been
left with minus a negative 40,
and we should know, double
negative rule says that
those do cancel out to a plus.
So same thing, just two
different ways to look at it.
Continuing on then, so the denominator,
what we would have, we
have that two times two,
which is a four, we can go
ahead and simplify that.
Okay, so I'm just gonna move this up here,
since I don't have much
room on the screen here,
and I wanna keep this going.
So I'm gonna bring over the
x is equal to negative three
plus or minus...
The only thing I'm gonna do here,
is underneath the square root symbol,
we have this nine plus 40,
well nine plus 40 is 49.
And we have that over four.
Okay, so at this point here,
we need to work on
simplifying down our radical.
So we have the square root of 49,
we need to take the square root of 49.
So sometimes if we're
lucky like we are here,
we are left with a perfect
square underneath that radical.
So remember, a perfect square
is when you have the same
number multiplied to
itself to get something.
So looking at 49, well what times what,
or what times itself will give you 49?
Well 49 happens to be seven times seven.
So the square root of 49 is seven.
So I'm gonna go ahead
and break that down next,
x is equal to negative three
plus or minus seven, over four.
Okay, so the only thing left
to do with this problem,
we're almost done, is to take care of this
plus and minus sign that
we have right there.
So when it does come to the
quadratic equation here,
right, that plus or minus
part of the formula,
and remember that that comes
from taking that square root.
Because, seven times seven is 49,
but isn't negative seven
times negative seven also 49?
Right, and we really don't
have a way of knowing which
one is it, is it the positive
ones, is it the negative ones?
So we actually assume that it's both.
So at this point in our problem,
what we do is we kinda split this up.
We're gonna take that negative three,
and then we're gonna say
plus seven over four,
and then we're gonna
take that negative three,
and we have minus seven over four.
And what we're gonna do, is
we're gonna simplify each one
of those, and we'll have two answers here.
So going ahead and simplifying here,
negative three plus seven is
a four, and that's over four.
And four divided by four is a one there.
So one is our first answer here.
Simplifying the next one,
negative three minus seven is
a negative ten over that four,
and we do need to simplify
down that fraction,
with ten over four, we
do have a two in common,
so we can divide out a two to be left with
a negative five halves.
So here we have two solutions,
and that's almost always gonna happen
when it comes to solving
quadratic equations.
That little exponent right here, this two,
typically means that you're
gonna have two solutions.
Had it been a cubed,
then we'd probably be
looking for three, and so on.
So that two signifies hey, we
could have two solutions here,
and our two solutions for
this problem is the one,
and a negative five halves.
Okay, so let's try working
on another example.
So here we're given
another quadratic equation
that we need to plug and chug
into the Quadratic Formula.
So fist thing hopefully
that you should be noticing
about this equation, that
it looks slightly different
than the other one.
Alright, so we have this
negative 8x term over there.
Now drawing our attention
back to the quadratic equation
I have over here on the
left side of the screen.
Alright, a quadratic equation
is in the form ax squared
plus bx plus c, and
it's set equal to zero.
Now when we're plugging and chugging
into the Quadratic Formula,
it does need to be in the proper format.
Our x squared term has to go first.
Our x to the first power
term should be next,
followed by our constant,
and it should all be set equal to zero.
If we're given an example
where that is not so,
that means we need to have a little bit
of a rearranging game going on,
we need to move some stuff around
so that it can be in the proper format.
So then if we wanna get
this into the proper format,
we need to move that negative 8x over,
so we would have to
add that to both sides.
So I'm gonna say plus 8x, plus 8x,
giving us, well I'm gonna drop down
the x squared term first,
then plus that 8x, then plus the one,
so it's in the proper form.
And then that's all going
to be set equal to zero.
Now that it's in the proper form,
now we can figure out what is a?
What is b?
And what is c?
Okay, so now remember,
a is always in front of our squared term,
so here's our x squared.
Remember if there's not a number there,
it is understood to be a one,
so a is one for this one here.
B is the coefficient of our
x to the first powered term,
so in front of our x we have an eight.
And c is our constant,
we have a one there.
So a is one, b is eight, c is one.
So let's plug and chug
this into the formula.
So we have x equals
negative b, our b is eight,
plus or minus the square root of
b squared, so that means eight squared,
minus four times one, times one,
all over 2a, so two times one.
Now let's go ahead and simplify this down,
so dropping down the x is equal to,
let's also drop down that negative eight
plus or minus the square root of...
Remember we wanna focus on
what's underneath the radical first.
Underneath the radical, the
first thing we have there
is eight squared, so
eight times eight is 64.
So remember, Order of Operations
tells us we need to take
care of exponents first.
After we take care of the exponent,
we have multiplication over here,
four times one, times one.
Well, four times one,
times one is just a four.
So scooch over the minus,
and that's a four there.
And that's all over two times
one, which is just a two.
Simplifying further, let's
go ahead and take care of
that subtraction,
so again I'm gonna just
move this all over,
x is equal to negative eight,
plus or minus the square root of,
well 64 minus four is 60,
and that's all over two.
So we need to work on
simplifying our radical there,
so over here off to the
side I'm gonna pull out
the square root of 60, because
we need to simplify that,
so we need to remember, how
do we simplify radicals?
Now we can't just plug and chug this
into the calculator and get a decimal,
that is not what they
are looking for here.
So when it comes to simplifying radicals,
remember we need to use perfect squares.
So perfect squares are the
four, nine, 16, 25, 36, 49,
and so on, right?
Two times two is four,
three times three is nine,
that's how you get those perfect squares.
And what you're looking for,
is which one of those
perfect squares can divide
into our number here,
or how can we take 60
and break it up into a
form of one of our factor,
using a perfect square.
So let's see, if we're looking at 60,
the first perfect square that we have
that's smaller than that is 49,
60 is not 49 times something.
36 would be the next one, but
60 is not 36 times anything.
It's not 25 times anything,
it's not 16 times anything,
it's not nine times anything.
Four would be the next one,
and actually four would work here
because four times 15 is 60.
So what we can do, the
product rule of radicals
says that we can take
the square root of 60,
and break it up to the square root of four
times the square root of 15.
Take the square root
of the one that we can,
the square root of four is a two,
drop down the other one.
So square root of 60 is
two square roots of 15.
Okay, so that means we have
x is equal to negative eight
plus or minus two square
roots of 15, over two.
Okay, so here we just have
a little bit of simplifying
left to do.
If we're taking a look at our problem,
we have three terms in this problem,
negative eight, two square roots of 15,
and this two down here.
Now we do have something in
common with all three terms,
eight, this eight here,
this two, and this two,
they do all have a two in common,
so we can simplify out,
divide out a factor of two from each one.
Or if we think back to dividing
a polynomial by a monomial,
for some people it's a little
bit easier to see it that way,
if they don't notice that
there's a common factor
of two here.
And this is what you would do.
So, dividing a polynomial
by a monomial says
that you take everything in the numerator,
and put it over the same denominator.
So we have a negative eight,
and put that over a two,
then we have plus or minus
the two square roots of 15,
and we put that over a two,
so we kinda split these up
into two different fractions.
Alright, the negative eight over the two,
then the two square
roots of 15 over the two.
And we simplify each one of these,
well, negative eight divided
by two is a negative four,
drop down the plus and minus sign.
And then here, these twos are
gonna cancel with each other,
remember any time you have
the same thing over itself,
they're gonna cancel.
So we can drop down
that square root of 15.
So depending on your instructor,
you may be able to state
this as your answer,
if not, if they do want it
split up into the two different
ones, then we would just say negative four
plus the square root of 15,
and then we have negative four minus
the square root of 15.
So these would be our two solutions
for this problem here.
Okay, so let's work on
one more example here.
Alright, so taking a look at this example,
it already is in the proper form, right?
We have all three of our
terms, it's set equal to zero.
Right, so, let's find what is a?
What is b?
And what is c?
So taking a look at our x squared term,
we have a two in front, a positive two.
So a is two.
Taking a look at our b to the first power,
b to the first power, excuse me,
taking a look at our x to
the first powered term,
we have a minus four in
front, so b is negative four.
And our constant here is a positive five,
so c is five there.
So let's plug and chug.
X is equal to negative b, so
that's negative, negative four,
plus or minus the square
root of b squared,
so that's negative four squared
minus four times a, times c.
And that's all over two times two, okay?
Simplifying this problem here,
well first off, so x is equal
to negative, negative four,
we can go ahead and cancel
out those negatives,
remember double negative rule
says that we can do that.
Drop down the plus, minus, square root of,
and taking care of our
exponent, negative four squared,
so negative four times
negative four is a 16,
minus four times two is eight,
eight times five is 40, and
this is all over two times two,
which is four.
Continuing on here, x is
equal to, move over the four,
plus or minus the square root of,
we need to do subtraction here.
16 minus 40, so remember
if you have 16 dollars
in your bank account,
and you spend 40 bucks,
well hey, you're now negative 24.
And that's all over four.
So I want you to pause
here for just a moment,
and take a look at this problem here.
Specifically focusing on this square root,
square root of negative 24.
So hopefully you have
learned at this point,
that we actually cannot
take the square root
of a negative number here.
We get, or our solution
is not a real number.
And this is why,
so remember what a square root implies,
so let me use the square root of 25.
Now, the square root of 25
is a plus or minus five.
It could either be a positive five,
because five times five is 25,
or it could be a negative five,
because negative five
times negative five is 25.
But, can you think of
a number times itself
that'll get you a negative,
a negative anything,
a negative 24, a negative, you know, 25,
there's really, honestly,
there's no possible way
that you can get this
negative inside here.
Because a positive times
a positive is positive,
and then a negative times a
negative, is also positive.
There's no way you can
multiply a number to itself,
to get a negative number.
So this is just not
possible, we can not do it.
We do not get a real number,
so our solution here, not a real number,
or no real solution, every book,
every teacher states that
just a slightly different way,
but there is no solution to this problem.
We cannot continue on, no solution.
