>> Hello, this is Professor Erbas White.
Today we will talk about quadratic equations,
primarily using the quadratic formula.
Make sure you watch the first video before you
do this to make a sense about this whole thing.
All right, I'll give you the quadratic formula.
The quadratic formula is a formula that is
used to solve quadratic equations of this form.
It needs to be memorized unless
you want to derive it on the spot
and I'll show you how to do that.
Here it is, negative b plus minus rad b square
minus 4ac over 2a, that needs to be memorized.
[Inaudible] how we derive it to start with.
We start with the quadratic equation.
Remember the complete in square method?
I'm going to follow those steps.
Move the constant term to the right, x
squared plus bx, move the c to the right,
divide everything by the leading coefficient,
divide everything by a. All
right, so far so good.
Let's move, now complete the square
on the left hand side of the equation.
So how did we do that?
You take this term, divided by
2 that would give you b over 2a
and then square it, b squared,
this is 4a squared.
Add that to both sides.
You use to have x squared plus b over
ax, add b squared over 4a squared.
And you had negative c over a.
Add b squared over 4a squared.
Now, this became the perfect
square of x plus b over 2a.
Now I need to put it in the same denominator.
What am I missing?
I'm missing 4a.
Rewrite, negative c, 4a plus
b squared over 4a squared.
Let's make this pretty, put this first,
b squared minus 4ac over 4a squared.
What do I do next?
You go, you take square root of both sides.
What do you end up with?
Absolute value of x plus b over 2a.
On the other side, take this out.
You're going to have b squared minus 4ac over.
This comes out as 2a.
All right, now let's take
out the absolute value.
You're going to end up with
plus minus x plus b over 2a.
You have b squared minus 4ac over 2a.
Move the plus minus on the other side.
Then move b over 2a on the other side.
Make it pretty.
Ta-da. Here's your quadratic formula.
All right, and we did this solve portion.
All right, now that we have our
quadratic formula, how do we use it?
I'll give you the steps and if you follow
the steps, you'll be in good shape.
The question is solve, the first thing using
the quadratic formula, write the abc's down.
A is this guy.
B is this guy.
C is this guy.
Next thing, write what's in this radical.
That's called a discriminent,
b squared minus 4ac.
B squared is 2 squared minus 4, a is 1, c is 1.
This one is 4 minus 4, that's zero, okay?
Now let's plug it in.
X equals, negative b is negative
2, plus or minus rad zero over 2a.
Rad zero is zero, so this
plus minus portion goes away,
negative 2 over 2, which is negative 1.
And actually, I do know with factoring this is -
the factor form is x plus 1 square,
that means x equals negative 1.
So this is consistent with that.
This happens to be an easy example.
All right, how do I do this one?
The question is solve, write down your abc's.
Write down your discriminent.
B squared is 16 minus 4, a is 2, c
is negative 5, 16 minus, minus, plus,
2 times 5, that's 10 times 4, 40, 56.
All right, now what do you have?
X equals negative b, which is negative
4, plus or minus rad 56 over 2a.
What is 7 times 7 is 49, 8
times 8 is 64, 7 times 8 is 56.
So let's go and write that, 7
times 8, 7 times 8 is 2 and 4.
All right, over 4.
What is that?
Negative 4, plus or minus,
4 comes out and you have 14.
If you clean this up, you're going to end
up with negative 1, plus or minus, 2 is 1/2.
You can do it that way.
All right, okay, well, I
use the quadratic formula.
Let's go and do another one.
The question is solve, a is
3, b is negative 4, c is 5.
Write down the discriminent.
B squared is 16, minus 4, a is 3, c is 5.
Okay. This one gives you 16 minus
3 times 4 is 12 times 5, that's 60.
That one is 60, 50, 44, right?
44, 56, yes, negative 44.
So what do I have?
X equals negative b, which is
4, plus minus rad, 44 over 2a,
which is 6, but this one is a negative.
All right, so that means I'm
going to get an i in there.
So 4 plus or minus i, 44
is 4 times 11, 2 comes out.
And i over 6, and let's get
rid of some of these guys.
This will give me 2/3 plus or minus, 2 over
6 that 1/3, rad 11, i. Now what did I end up?
A complex pair, so let's summarize all this.
When do I get what?
It looks like everything depends on what
is happening with b square minus 4ac.
I did all three cases.
When did b square minus 4ac was negative,
I ended up with a complex
pair no real answer, okay?
When b square minus 4ac was zero, remember that
discriminent part dropped off of the equation,
you ended up with one real answer.
When b square minus 4ac was positive,
I ended up with two real answers.
Now, you have two different real of answers.
It was a nice answer when you didn't have
irrationals, you either had two rational reals
when the discriminent is perfect square.
When the discriminent was ugly
like 7, 11, those kind of things,
you ended up with irrational real answers
and that means the discriminent
is not a perfect square.
So this pretty much summarizes it.
You either get a complex pair, one real
answer, two real answers and then one
of the two real answers you either have
two rational or two irrational answers.
