Welcome to part two of the presentation on
quadratic equations.
Well, I think I thoroughly confused you the last time
around, so let me see if I can fix that a bit by doing
several more examples.
So let's just start with a review of what the
quadratic equation is.
The quadratic equation says, if I'm trying to solve the
equation Ax squared plus Bx plus C equals 0, then the
solution or the solutions because there's usually two
times that it intersects the x-axis, or two solutions for
this equation is x equals minus B plus or minus the square root
of B squared minus 4 times A times C.
And all of that over 2A.
So let's do a problem and hopefully this should make
a little more sense.
That's a 2 on the bottom.
So let's say I had the equation minus 9x squared minus
9x plus 6 equals 0.
So in this example what's A?
Well, A is the coefficient on the x squared term.
The x squared term is here, the coefficient is minus 9.
So let's write that.
A equals minus 9.
What does B equal?
B is the coefficient on the x term, so that's this term here.
So B is also equal to minus 9.
And C is the constant term, which in this example is 6.
So C is equal to 6.
Now we just substitute these values into the actual
quadratic equation.
So negative B, so it's negative times negative 9.
That's B.
Plus or minus the square root of B squared, so that's 81.
Right?
Negative 9 squared.
Minus 4 times negative 9.
That's A.
Times C, which is 6.
And all of that over 2 times negative 9, which
is minus 18, right?
2 times negative 9-- 2A.
Let's try to simplify this up here.
Well, negative negative 9, that's positive 9.
Plus or minus the square root of 81.
Let's see.
This is negative 4 times negative 9.
Negative 4 times negative 9 is positive 36.
And then positive 36 times 6 is-- let's see.
30 times 6 is 180.
And then 180 plus another 36 is 216.
Plus 216, is that right?
180 plus 36 is 216.
All of that over 2A.
2A we already said is minus 19.
So we simplify that more.
That's 9 plus or minus the square root 81 plus 216.
That's 80 plus 217.
That's 297.
And all of that over minus 18.
Now, this is actually-- the hardest part with the quadratic
equation is oftentimes just simplifying this expression.
We have to figure out if we can simplify this radical.
Well, let's see.
One way to figure out if a number is divisible by 9 is to
actually add up the digits and see if the digits
are divisible by 9.
In this case, it is.
2 plus 9 plus 7 is equal to 18.
So let's see how many times 9 goes into that.
I'll do it on the side here; I don't want to be too messy.
9 goes into 2 97.
3 times 27.
27-- it goes 33 times, right?
So this is the same thing as 9 plus or minus the square root
of 9 times 33 over minus 18.
And 9 is a perfect square.
That's why I actually wanted to see if 9 would work because
that's the only way I could get it out of the radical, if
it's a perfect square.
As you learned in that exponent rules number one module.
So this is equal to 9 plus or minus 3 times the square
root of 33, and all of that over minus 18.
We're almost done.
We can actually simplify it because 9, 3, and minus 18
are all divisible by 3.
Let's divide everything by 3.
3 plus or minus the square root of 33 over minus 6.
And we are done.
So as you see, the hardest thing with the quadratic
equation is often just simplifying the expression.
But what we've said, I know you might have lost track-- we did
all this math-- is we said, this equation: minus 9x
squared minus 9x plus 6.
Now we found two x values that would satisfy this equation
and make it equal to 0.
One x value is x equals 3 plus the square root
of 33 over minus 6.
And the second value is 3 minus the square root
of 33 over minus 6.
And you might want to think about why we have
that plus or minus.
We have that plus or minus because a square root could
actually be a positive or a negative number.
Let's do another problem.
Hopefully this one will be a little bit simpler.
Let's say I wanted to solve minus 8x squared
plus 5x plus 9.
Now I'm going to assume that you've memorized the quadratic
equation because that's something you should do.
Or you should write it down on a piece of paper.
But the quadratic equation is negative B-- So b is 5, right?
We're trying to solve that equal to 0, so negative B.
So negative 5, plus or minus the square root of B squared-
that's 5 squared, 25.
Minus 4 times A, which is minus 8.
Times C, which is 9.
And all of that over 2 times A.
Well, A is minus 8, so all of that is over minus 16.
So let's simplify this expression up here.
Well, that's equal to minus 5 plus or minus
the square root of 25.
Let's see.
4 times 8 is 32 and the negatives cancel out, so
that's positive 32 times 9.
Positive 32 times 9, let's see.
30 times 9 is 270.
It's 288.
I think.
Right?
288.
We have all of that over minus 16.
Now simplify it more.
Minus 5 plus or minus the square root-- 25 plus
288 is 313 I believe.
And all of that over minus 16.
And I think, I'm not 100% sure, although I'm pretty sure.
I haven't checked it.
That 313 can't be factored into a product of a perfect
square and another number.
In fact, it actually might be a prime number.
That's something that you might want to check out.
So if that is the case and we've got it in completely
simplified form, and we say there are two solutions, two
x values that will make this equation true.
One of them is x is equal to minus 5 plus the square
root of 313 over minus 16.
And the other one is x is equal to minus 5 minus the square
root of 313 over minus 16.
Hopefully those two examples will give you a good
sense of how to use the quadratic equation.
I might add some more modules.
And then, once you master this, I'll actually teach you how to
solve quadratic equations when you actually get a negative
number under the radical.
Very interesting.
Anyway, I hope you can do the module now and maybe I'll add a
few more presentations because this isn't the easiest module.
But I hope you have fun.
Bye.