Welcome.
So what I'd like
to do is show you
how to solve a
quadratic equation.
Solve a quadratic equation by
using the quadratic formula.
So what I did is I did a
nice little coloring scheme.
And it takes a while,
so I didn't even
do the discriminant like that.
But basically the
main important thing
I want to do in solving the
quadratic equation, obviously
the best way to be able to look
at this, is by using factoring.
However, if you can't
solve a quadratic equation
by factoring, then
we could always
use the quadratic
formula-- where
the solutions, what
the value is of x
when equation is
equal to 0 are going
to be by using this format.
Now the way I get
a, b, and c from
is from the quadratic equation
when it's set equal to 0.
a, b, and c.
So the main important
thing, the thing
that I want you to
understand most,
is the quadratic formula works
when your quadratic equation is
set equal to 0.
So for this example, you can
see it's not set equal to 0.
So the first thing we want to
do is add a 6x onto both sides
and add a 1 onto both sides.
Therefore I have the
equation 9x squared
plus 6x plus 1 equals 0.
Then the next
thing I like to do,
is to identify what is
your a, b, and your c.
Well, remember a is going
to be your coefficient
of your quadratic term.
b is your coefficient
of your linear term.
And c is your value
of your constant.
So a in this case is 9.
b in this case 6.
And c is going to be equal to 1.
Now the next thing I
think is very helpful
is, rather than
plugging in everything
into this formula,
let's figure out what
the discriminant is first.
Because the discriminant
is all the values
that are under the square root.
And knowing what type
of number that is
can be very, very helpful.
If that number is 0,
then that tells us we're
just going to have one solution.
If that number is
a rational number--
a real rational
number-- therefore
then we know that
there's going to be
two possible rational
zeros or I'm sorry.
If that number is
a square number,
then we'll have two
rational real solutions.
If that number is a
non-square number--
like a square number would
be 9, a non-square number
would be 5-- if it's
a non square number,
then we're going to have two
real irrational solutions.
And if that number is
negative, then we're
going to have two imaginary,
or complex solutions.
So I always like to go ahead
and figure out the discriminant
first.
And by doing the
discriminant-- now
that's not part of
this problem, but there
are questions that ask what
are the types of solutions--
So I'm not going to write it
down, because that's not really
what we're doing.
But I will describe
what the solution is.
OK.
So therefore I have 36 minus
4 times 9, which is 36.
So that equals the square root
of 0, which is equal to 0.
So since we have
a vector 0, we're
going to have one real solution.
OK?
Now up here, see what happens.
Let's see how that works.
Let's just pretend
the square root of 0.
Let's plug that back in with
the rest of the equation.
So my solutions are going to
be x equals opposite of b.
So b in this case is 6, right?
It's not negative
6 from up here.
But b is actually 6 when
I get it set equal to 0.
So the opposite of b is
negative 6 plus or minus
the square root of 0,
which we know is 0.
But I'll just write it in there.
Divided by 2 times a,
which in this case is 9.
So negative 6 plus or
minus the square root of 0.
That's just going to equal-- x
equals negative 6 plus or minus
0 is just going to be
negative 6 over 2 times 9
is going to be 18.
And therefore that
equals a negative 1/3.
So therefore my one
solution, the value
where the graph
crosses the x-axis,
is going to be negative 1/3.
In the next example over here.
We're going to do the
exact same case here.
First thing we need do is
identify our a, b, and our c.
a in this case is 1.
b in this case is 4.
And c in this case is
going to be a negative 3.
Again let's determine
the discriminant first.
So the discriminant is going to
be the square root of b squared
minus 4 times a times c.
So therefore in this case
I'm going to have 16 minus,
or negative 4 times
negative 3 times 1
is going to be a positive 12.
So the discriminant equals
the square root 16 plus 12,
is going to be 28.
We can break apart
the square root of 28
into the square
root of 4 times 7,
which is equal to
2 square root of 7.
Now 28 is not a square
number, you can see.
So therefore our
solutions are going
to be two real
irrational solutions.
So now, let's go and
find the solutions here.
So I have opposite of b.
So it's going to be
negative 4 plus or minus
my discriminant here, which
is 2 square root of 7.
Divided by 2 times
a, which is 1.
So that's 2 times 1, is just 2.
Now I see I can divide
this 2 into my negative 4.
And divide the 2 into
the 2 square root of 7.
So my final answer is negative
2 plus or minus the square root
of 7, because that 2
divides into both of them.
2 divided by 2 is 1.
1 times the square root of 7.
So it's really
helpful to determine
what the discriminant is.
Not only does that tell you
what type of solutions you have,
but it also lightens
up your workload.
OK.
In this example, you can see
that, obviously, my equation
is not set equal to 0.
So that's the first
thing I'm going to do.
I'm going to add
3 to both sides.
So therefore I have 6x squared
minus 8x plus 3 equals 0.
Now I can say that a equals
6, b equals negative 8,
and c equals 3.
Now all the determine
my discriminate first.
So I do square root of b
squared minus 4 times a times c.
OK.
So 8 squared is going
to leave me with a 64.
Minus 4 times 6
is going to be 24.
Times 3 is going to
be times 3, 24, 48.
That's going to be 62.
Right?
24 times 3, 24, 48.
No, that's going to be 72.
2, 4, 8, 12 So that's
going to be 72.
So now when I go ahead
and simplify that,
that gives me the square
root of negative 8.
So therefore since I'm
taking the square root
of a negative number,
I'm going to now have
two complex solutions.
Meaning the graph
is actually not
going to cross the y-axis
at any real values.
However I still
can simplify this.
I can break this down--
Let's do that over here.
I can break this down
the however to 4 times
2 times negative 1.
Well, the square root
of 4 is going to be 2.
Times the square root of 2.
And the square root
of negative 1 is i.
In case you're used to
using your imaginary units.
If not, you're going to
say no real solutions.
So let's go ahead
and figure out what
these complex solutions are.
So opposite of 8.
Now in this case
my 8 is negative.
So now it's going to be
positive 8 plus or minus
my new discriminant, which is 2.
Times the square root of 2, i.
All over 2 times a, which is 12.
2 times a, which is 6.
So that's going to be 8 plus
or minus 2 square root of 2i.
All over 12.
Now you can see out of all of
these I can factor out a 2.
Or I can divide out a 2 here.
2/12 is 1/6 and therefore
2 that would be a 4.
So my final answer
is going to be
4 plus or minus the square
root of 2i divided by 6.
OK.
Or if you're not
used to i and you're
like, what the heck is i?
You just say no real solutions.
OK.
Last one we're
going to get into.
Here I have 5 times x
squared plus 2x plus 5.
So in this case, again, we're
going to identify a equals 5.
b equals 2.
c equals 5.
And to determine my
discriminant here,
I'm going to do b squared.
So it's going to be 2 squared
minus 4 times a times c.
So that's going to be 4
discriminant-- actually
I don't want to do that.
Let me change this problem.
Let's see here, I'm going
to do x squared minus 4
times a times c.
So that's going to be 4.
4 times a.
1 times c.
4x plus 2.
Oh 4x plus 5 equals 0.
OK.
So let's go ahead
and do this one here.
I want to give you one
example of one solution when
you have irrational,
when you have complex,
and then one where you're
going to have two rationals.
So again let's do
this one, a equals 1.
b equals negative 4.
c equals 5.
The discriminant
here is going to be
the square root of
negative 4 squared
minus 4 times a, which is 1.
Times c, which is 5.
b squared minus 4
times a times c.
Negative 5.
Oh that should be
negative, that's right.
I did my math wrong.
That should be times negative 5.
The problem is x squared
minus 4x minus 5.
I was doing my factoring
in my head wrong.
OK.
So therefore, I now have
the square root of 16.
And then negative
4 times negative 5
is going to be a
positive 20, which
equals the square root
of 36, which equals 6.
Now to solve rest it.
x equals the opposite
of b, which is going to be 4.
Plus or minus 6, which
is my instrument.
All over 2 times
a, which is just 2.
So now, I can do 4
plus 6, which is 10.
10 divided by 2 is
going to equal 5.
And then 4 minus 6,
which is negative 2.
Divided by 2 is negative 1.
So there we go,
ladies and gentlemen.
That is how you solve a
quadratic equation using
the quadratic formula.
Thanks.
