>> A quadratic equation in general form
is ax squared plus bx plus c equals 0.
So, that might be 2x squared plus 5x minus 3
equals 0, so the coefficients are your values
of a, b and c. To be a quadratic a
can't be 0, but b and c can both be 0.
So, this is a quadratic formula and that
gives the solutions for x in terms of a,
b and c and later in this video I am
going to show you how you would come
across that formula, how we derive it.
But, the quadratic formula here says x
equals negative b or the opposite of b
or some people even say minus b, which is not
technically correct, but you get the idea.
So, negative b plus or minus the square root
of b squared minus 4ac and it is all over 2a.
Well, you could learn this formula
by singing a little song to "Pop!
Goes the Weasel," so here goes.
Here is what is you are going to sing
down here: x equals negative b plus
or minus the square root of b
squared minus 4ac, all over 2a.
Now, there is more to it, but
I am going to leave it for now.
So, this is the main one, so
sing it with me, you ready?
x equals negative b plus or minus the square
root of b squared minus 4ac, all over 2a.
Okay, so listen, sing it about
ten times for five days in a row
and you are never going to forget this song.
Now, are you ready to see where -- how
come that is true, why is that the solution
to a quadratic equation in general form.
You probably aren't that
interested but maybe you are.
So, those of you who are just
keep on going to the next part.
Alright, we are going to try and define
the solutions of ax squared plus bx plus c
and we were going to be doing
completing the squares.
So, you will have to have had some
instruction on that to fully understand this.
Alright, so step one is I have
got something in standard form,
so it is equal to 0, it is an equation.
I am going to start off by dividing both sides
by the coefficient of x squared which is a,
so divide everything by a. Now, notice I
put each term over a. It is really the same
as putting it all over a, but then of
course you would break it up -- oops.
And you could put each term over a, right?
So, we would have ax squared over a plus
bx over a, which I am going to write as b
over a times x, okay, plus c
over a, and 0 divided by a, 0.
When 0 is in the numerator you are going to
have 0 and if you remember I said a can't be 0,
the coefficient of x squared, so, we don't
have to worry about that we are dividing by 0.
Alright, so this gives me
x squared plus b over ax.
Now, I am going to take the constant term
c over a and subtract ca from both sides.
So, I am going to put on the other
side as negative c over a. I am going
to leave a little space here, maybe closer
to negative c over a. And I am going
to try to complete the square here.
I am going to add something to
both sides of the equation, okay?
Now, what is it going to be?
Well, if this was a perfect square then to
get an x squared here it is going to have
to be x times x to get the x squared
so that is an x. Since this is a plus,
this middle term is a plus and then hopefully
you remember this term in here you are going
to multiply it by 2 to get b over a. So, it
is actually half of b over a to begin with.
So, it is going to be half of the coefficient
of x. So, half of b over a is b over 2a.
So, let's see if that would work.
If I am going to use my formula for squaring a
binomial look down here I would get x squared
so x times x is x squared and then I would have
for the middle term remember it is x times b
over a, I mean x times b over 2a times 2.
So, convince yourself if you did, x, let's see.
Let's move this up and what
is x times b over 2a times 2.
To see all the 2's and
lo and behold I get b over ax.
So, that is correct for the middle term and
then the last term which is what we really want
to know what goes in the box
here is the last term squared.
So, b over 2a times b over 2a
is b squared over 4a squared.
Whatever I add to the left side of the equation
I must add to the right side of the equation.
Okay, so I have b squared over 4a squared
minus c over a. Oh, but how we are going
to simplify on the right hand side?
I have some fractions.
So, I need to get a common denominator and that
common denominator is going to be 4a squared.
So, this has to be multiplied
by 4a over 4a, let's see.
Okay, hopefully you could see all that.
Alright, so what do we have?
Keep on going.
So, I have got this x plus b over 2a squared
and I have a common denominator, right,
b squared over 4a squared minus -- I am
going to write this as 4ac just write it
in alphabetical order over 4a squared.
So, I have x plus b over 2a squared is
b squared minus 4ac over 4a squared.
Okay, now so I know this thing squared
is what's over here and now to simplify,
I mean to figure out what x is I am going
to take the square root of both sides.
[ Pause ]
>> Okay, so when you take the
square root of both sides...
[ Pause ]
>> Remember you have to put a plus or
minus in front of one of the sides.
[ Pause ]
>> Alright, so if I take the square root
of something squared I get what is
inside which is x plus b over 2a.
And on this side I have a plus or
minus the square root of the top
which is b squared minus 4ac over
the square root of the bottom.
Well, the square root of 4a
squared is 2a, almost done.
So, now I am going to subtract
b over 2a from both sides.
So, I get negative b over 2a and I have
this plus or minus b squared minus 4ac
over 2a and great, common denominator.
So, I have it all over 2a.
We have negative b plus or minus the square
root of b squared minus 4ac, all over 2a.
Awesome! There it is the quadratic formula.
Sing it one more time, x equals
negative b plus or minus the square root
of b squared minus 4ac, all over 2a.
Hey, if you could derive the quadratic formula,
wow you really understand completing the square.
But if you can't derive it at least
you got to know formula and use it.
Let's just do one little
problem using the formula.
So, here is a problem.
What is a, b and c. Well, a is
the coefficient of x squared,
b is the coefficient of x, and c is negative 7.
I like to do b squared minus 4ac
first that is called the discriminate,
so b squared is negative 3 times
negative 3 it is 9 minus 4ac.
Alright, I am going to do a times c,
2 times negative 7 is negative 14.
So, I have 9 plus 56 which is 65.
Alright, so I know when I plug in for b squared
minus 4ac I have already done that part.
So, now let's use the quadratic formula it's x
equals negative b that means the opposite of b,
3 plus or minus the square root of 65,
I already did the b squared minus 4ac,
all over 2a. A is 2 and there you go.
La de da de da.
Two solutions, 3 plus the square root of 65 all
over 4 and 3 minus the square root of 65
over 4, these are really two solutions.
Okay, so we have got the quadratic
formula, how I derive the quadratic formula,
how to sing it so you could memorize
it, and just one little example.
Have fun!
