- [Instructor] Hello,
today I will solve by using
the quadratic formula.
I will just be demonstrating how
to use this formula right over here
on the right hand side,
the quadratic formula.
You want to have this memorized
and I will be demonstrating with,
two x squared, minus 15x,
minus 27 equal to zero.
So we have this equation,
now it is in the form ax
squared, plus bx, plus c
equal to zero.
And when we're in this form, that tells us
we can use the quadratic formula to solve.
The quadratic formula will always work,
if you're in this form no matter what.
So, now I need to figure out what is my a,
what does my b and what is my c?
That'll be the first step.
Well a, a is the leading
coefficient on the x squared value.
So a is going to be two,
b, is going to be the
leading coefficient on x.
As you can see here, it's negative 15.
And then our c is just the
last constant, is negative 27.
Now I'm gonna use my a, b and c
and plug it into my quadratic formula.
So let's start,
we have x equals negative b,
so the b was what I had in green here.
So I'm gonna put negative 15
and then we had plus minus the
square root of all of this.
So we have b squared and
you see here I am putting
everything in parentheses
that I'm plugging in.
And I'm doing this so
that I can clearly see
when the negatives become squared
or when you have a
negative times a negative
as we do in both of these cases.
If you do not do that,
then you'll be missing
signs that you need,
whether it's positive or negative.
And then we have minus
four times a times c.
Now a, I put in yellow, which was two
and then c was negative 27.
And then in the denominator,
we have two times a, and a,
we've labeled with yellow,
so we have two times two.
Now we need to just keep cleaning up.
Now we see in this very
first part right here
we have a negative times negative
that would give you a positive 15.
Okay, negative 15 all
squared, that is 225.
And we have negative four
times two times negative 27
Now the negative and the negative,
this negative and this
negative would really make
a positive.
Now four times two is eight,
eight times 27, that's a large number,
I recommend using your
calculator, that is actually 216.
And I'll divide this all by
two a, which is two times two,
it's all divided by four, alright?
So we have set up entirely everything
and we're right on our way
and we actually don't
have too much left to do.
I'm gonna keep all of this
in the same form, that is,
it's all divided by four,
we've 15 plus minus the square root piece.
Now 225 plus 216 actually gives you 441.
Now what do we do?
Well, 441 we're taking
the square root of this.
We need to clean that up.
So what I'm gonna to do,
is I'm gonna move over to the side.
I'm just gonna go right over here
and think about the square root of 441.
Well, let's make a factor tree,
441, now, I'm gonna divide
it by three, just to,
so you take 441 divided
by three you get 147.
Now this is odd, so let's
try dividing by three again,
if I divide that by three, you get 49.
Now 49 you know that that
is seven times seven,
so I have three, three, seven and seven.
So I know, based on what you
know about radical rules,
you could write this as three times three
times seven times seven.
And when you take the
square root of anything,
well, you're looking for pairs of threes.
So pair of three can come out,
then we have times a pair
of sevens can come out.
So this will just leave us with 21.
So the square root of 441
equals 21, let's check this.
The way we could check that,
well, this is saying that
it's a perfect square.
So 21 squared, is what?
Well, 21 squared, if you put
it in your calculator, is 441.
So that is a fact.
If you have the square root
of 441, you'll have 21.
So I'm gonna go back over here.
We have x equals 15 plus or minus 21
and this is all divided by four.
Now what we have here
is a plus and a minus.
So let's split this up into two cases.
We'll have x equals 15 plus 21 over four,
and then we'll have x equals
15 minus 21 over four.
This is just to help us
with keeping organized,
because we have two solutions truly.
Now what I'll do is I'll
go down a little bit,
and just say x is equal
to, now if we do 15 plus 21
that would bring us to 36 over four,
and 36 over four would
give you x equals nine.
And we have x equals 15 minus 21,
so we know we'll get a negative,
15 minus 21 gives you
negative six divided by four.
Now what does six and four have in common?
Well, six and four have got two in common,
and since two is in common,
we'll have negative three over two.
We've simplified the fraction.
So we have a solution x equals nine,
we also have a solution x
equals negative three halves.
Either of these, would make
the original equation equal to zero.
It will solve this equation,
that we started with
at the very beginning.
