Okay. So, in this video we're going to talk about what's called
the quadratic formula.
So, in our quest to solve quadratic equations, we have used factoring, we've
used solving by completing the square, we've used the square root method.
And it turns out that if you solve by completing the square generally, you can
get what's called the quadratic formula.
Now, the important thing about the quadratic formula is that it can be used to
solve any quadratic equation when the equation's been written in standard form.
And remember what we mean by a quadratic equation in standard form, it's this.
It's a -- let's rewrite that.
It's ax squared -- ax squared plus bx plus c is equal to 0; okay?
That's what we mean by standard form for a quadratic equation.
Now, just as a historical note, notice this is a polynomial equation where the
polynomials have degree 2.
So if you think about second degree equations -- so second degree equations --
the quadratic formula that I'm going to present in a moment was written down in
its most general form about 500 years ago.
So the formula I'm going to present to you, somebody wrote that down most
generally about 500 years ago.
And if this was 3rd degree, right, that if you consider not a quadratic equation
but a cubic equation, there's a formula for that as well, and that was
discovered about 450 years ago.
And you might ask yourself what about a 4th degree, and the answer is yeah, just
kind of about the same time.
So also about 450 years ago somebody wrote down a general formula for a 4th
degree polynomial equation.
And then you say, okay, what about a 5th degree?
And actually there is no such equation.
So what that means is that not just for 5th degree but whenever the exponent
gets to be a 5 or higher, there's no such equation that somebody can write down
that will give you the general solutions to that equation.
That was shown about 200 years ago.
That's kind of a miracle.
Think about that.
Somebody -- you know, for the first few examples, they showed that such an
equation does exist, right, up to 4th degree.
But for 5th degree and higher, somebody showed that something doesn't exist.
That's much more difficult, right, to show the non-existence of something.
And that's a really interesting story, and mathematically really beautiful.
It's really elegant, the theory that addresses that issue.
So the quest to find a general formula has been one that sort
of spanned the ages.
And the quadratic formula, that is for a quadratic equation, here's the formula
that gives you the solutions.
It's this, it's x is equal to negative b plus or minus the square root of b
squared minus 4 ac, and then all over, you have to divide that whole thing by 2
times a; okay?
This is called the quadratic formula.
Okay. Let's give that a box.
So this formula, the quadratic formula, like I was saying, was discovered,
written down generally about 500 years ago.
Let's go ahead and see how that helps us to solve any quadratic equation.
Let's take some examples.
Let's solve 2x times x minus 2 is equal to 3.
So the first thing we want to do is write this in standard form, ax squared plus
bx plus c is equal to 0.
So I'm going to take this 2x and I'm going to distribute.
So 2x times x is 2x squared.
And then 2x times negative 2 is negative 4x.
And then it is equal to positive 3.
So we want 0 on the right-hand side.
So it's going to be 2x squared minus 4x minus 3 is equal to 0.
At this stage, you want to identify your a, your b, and your c, because the
formula references those values; right?
So let's write that down.
A is equal to, well, for this problem a is 2.
B is actually negative 4.
And c is negative 3.
Now, the quadratic formula says that x is equal to negative b plus or minus the
square root of b squared minus 4 ac, and then all over 2a.
So what we're going to do is we're going to plug in our values of a, b, and c
into this formula.
So we've got, let's see, negative b.
Well, b is negative; right?
So it's negative negative 4, and that makes it positive 4.
Do you see that?
It's the negative of b, but b is negative, so the negative of a
negative is positive 4.
So plus or minus the square root of b squared, that's going to be 16, minus 4.
And I like to multiply a and c together, and when I do that, a times c
is negative 6.
I'm going to put that in parentheses because I'm going to multiply that by the 4
that's there in a minute.
Okay, all over 2 times a, which is 4.
So it's going to be x is equal to 4 plus or minus the square root of, well it's
going to be 16 plus 24.
I multiplied negative 4 times negative 6.
And then all over 4.
So let's finish this up here now.
So I've got x is equal to 4 plus or minus the square root of, well, 40 over 4.
Now, the square root of 40 can be simplified.
So it's going to be x equals 4 plus or minus the square root of 4 times 10,
all over 4.
So x is equal to 4, plus or minus, and the square root of 4 is 2, so 2 root
10, over 4.
Now, remember, you can't just cancel these right away, and the reason is because
it's addition right here.
Addition or subtraction, you have to factor before you can cancel.
So I'm going to factor a 2 out of both of these.
So it's going to be x is equal to 2 times 2 plus or minus the square root of
10 over 4.
And then finally I can cancel.
So this 2 cancels with a factor of 2 there.
And my answers are 2 plus or minus the square root of 10 over 2.
I have two answers, right, 2 plus root 10 over 2, and 2 minus root 10 over 2.
So let's go ahead and take another example using the quadratic formula.
Let's solve x squared plus 2x plus 3 is equal to 0.
And again, identify your a, b, and your c.
So a is equal to 1, b is equal to 2, and c is equal to 3.
So the quadratic formula says x is equal to, okay, negative b.
So it's negative 2 plus or minus the square root of b squared, which is
4, minus 4.
And a times c is going to be positive 3, right, a is 1, c is 3, a times c is 3.
Then all over 2 times a, which is just 2.
So x is equal to, let's see, it's going to be negative 2 plus or minus the
square root of, okay, 4 minus 12 under the radical, all over 2.
So x is equal to negative 2 plus or minus the square root of negative 8 over 2.
So x is equal to negative 2 plus or minus.
Now, the square root of a negative, that's going to be i, right, our buddy.
Square root of 8 all over 2.
Now, the square root of 8 can be simplified.
So let's finish it up here.
So x is equal to negative 2 plus or minus i times the square root of 4 times 2,
all over 2.
So x is equal to negative 2 plus or minus, and the square root of 4 is 2, so
it's 2i root 2, again, all over 2.
Now you have to factor so you can cancel.
So it's going to be 2 times negative 1 plus or minus i root 2, all over 2.
I factored a 2 out of both of these; right?
And the reason is because now I can cancel -- the twos cancel.
So here are my solutions, negative 1 plus or minus i square root of 2.
So I have two solutions, right, negative 1 plus i root 2, and negative 1
minus i root 2.
So notice in these problems that I was simply asked to solve; okay?
And I jumped right to the quadratic formula, but sometimes that's unnecessary.
Sometimes you find that you don't quite need the quadratic formula.
And let me give you an example of that.
Let's solve 2x squared plus 5x minus 3 is equal to 0.
Now, I can use the quadratic formula if I want, but in this problem it's much
easier to just factor.
So this is going to be 2x times, let's see, x is equal to 0; right?
So it's going to be this binomial multiplied by that binomial.
And it's going to be a positive 3 and a negative 1.
And so when I set these individually to 0, I get 2x minus 1 is 0, or x plus 3 is
equal to 0.
So this gives me 2x is equal to 1, so x is one-half.
That's one solution.
Or x is equal to negative 3.
So, in this case, I solve the equation without needing the quadratic formula.
And so you might ask yourself, well, how do I know if I need the quadratic
formula or if I can just simply solve by factoring?
And the answer is you always try to solve by factoring first.
And if it takes you longer than a couple minutes, then you move on to the
quadratic formula.
So, generally, solving by factoring is faster, but you don't want to waste too
much time on it.
If you're having trouble factoring, go ahead and jump right to the
quadratic formula.
So we're going to see how to use -- or why solving quadratic equations is
important when we craft quadratic functions.
Until next time.
