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Welcome to the presentation on
using the quadratic equation. So the quadratic equation,
it sounds like something very complicated. And when you actually first see
the quadratic equation, you'll say, well, not only does it
sound like something complicated, but it is
something complicated. But hopefully you'll see,
over the course of this presentation, that it's
actually not hard to use. And in a future presentation
I'll actually show you how it was derived. So, in general, you've already
learned how to factor a second degree equation. You've learned that if I
had, say, x squared minus x, minus 6, equals 0. If I had this equation. x
squared minus x minus x equals zero, that you could factor
that as x minus 3 and x plus 2 equals 0. Which either means that
x minus 3 equals 0 or x plus 2 equals 0. So x minus 3 equals 0
or x plus 2 equals 0. So, x equals 3 or negative 2. And, a graphical representation
of this would be, if I had the function f of x is equal to
x squared minus x minus 6. So this axis is
the f of x axis. You might be more familiar with
the y axis, and for the purpose of this type of problem,
it doesn't matter. And this is the x axis. And if I were to graph this
equation, x squared minus x, minus 6, it would look
something like this. A bit like -- this is f
of x equals minus 6. And the graph will kind of
do something like this. Go up, it will keep going
up in that direction. And know it goes through minus
6, because when x equals 0, f of x is equal to minus 6. So I know it goes
through this point. And I know that when f of x is
equal to 0, so f of x is equal to 0 along the x axis, right? Because this is 1. This is 0. This is negative 1. So this is where f of x
is equal to 0, along this x axis, right? And we know it equals 0 at the
points x is equal to 3 and x is equal to minus 2. That's actually what
we solved here. Maybe when we were doing the
factoring problems we didn't realize graphically
what we were doing. But if we said that f of x is
equal to this function, we're setting that equal to 0. So we're saying this
function, when does this function equal 0? When is it equal to 0? Well, it's equal to 0 at
these points, right? Because this is where
f of x is equal to 0. And then what we were doing
when we solved this by factoring is, we figured out,
the x values that made f of x equal to 0, which is
these two points. And, just a little terminology,
these are also called the zeroes, or the
roots, of f of x. Let's review that a little bit. So, if I had something like f
of x is equal to x squared plus 4x plus 4, and I asked you,
where are the zeroes, or the roots, of f of x. That's the same thing as
saying, where does f of x interject intersect the x axis? And it intersects the
x axis when f of x is equal to 0, right? If you think about the
graph I had just drawn. So, let's say if f of x is
equal to 0, then we could just say, 0 is equal to x
squared plus 4x plus 4. And we know, we could just
factor that, that's x plus 2 times x plus 2. And we know that it's equal
to 0 at x equals minus 2. x equals minus 2. Well, that's a little
-- x equals minus 2. So now, we know how to find
the 0's when the the actual equation is easy to factor. But let's do a situation where
the equation is actually not so easy to factor. Let's say we had f of x
is equal to minus 10x squared minus 9x plus 1. Well, when I look at this, even
if I were to divide it by 10 I would get some fractions here. And it's very hard to imagine
factoring this quadratic. And that's what's actually
called a quadratic equation, or this second degree polynomial. But let's set it -- So we're
trying to solve this. Because we want to find
out when it equals 0. Minus 10x squared
minus 9x plus 1. We want to find out what
x values make this equation equal to zero. And here we can use a tool
called a quadratic equation. And now I'm going to give you
one of the few things in math that's probably a good
idea to memorize. The quadratic equation says
that the roots of a quadratic are equal to -- and let's say
that the quadratic equation is a x squared plus b
x plus c equals 0. So, in this example,
a is minus 10. b is minus 9, and c is 1. The formula is the roots x
equals negative b plus or minus the square root of b squared
minus 4 times a times c, all of that over 2a. I know that looks complicated,
but the more you use it, you'll see it's actually not that bad. And this is a good
idea to memorize. So let's apply the quadratic
equation to this equation that we just wrote down. So, I just said -- and look,
the a is just the coefficient on the x term, right? a is the coefficient on
the x squared term. b is the coefficient on the x
term, and c is the constant. So let's apply it
tot this equation. What's b? Well, b is negative 9. We could see here. b is negative 9, a
is negative 10. c is 1. Right? So if b is negative 9 -- so
let's say, that's negative 9. Plus or minus the square
root of negative 9 squared. Well, that's 81. Minus 4 times a. a is minus 10. Minus 10 times c, which is 1. I know this is messy,
but hopefully you're understanding it. And all of that over 2 times a. Well, a is minus 10, so
2 times a is minus 20. So let's simplify that. Negative times negative
9, that's positive 9. Plus or minus the
square root of 81. We have a negative 4
times a negative 10. This is a minus 10. I know it's very messy,
I really apologize for that, times 1. So negative 4 times negative
10 is 40, positive 40. Positive 40. And then we have all of
that over negative 20. Well, 81 plus 40 is 121. So this is 9 plus or
minus the square root of 121 over minus 20. Square root of 121 is 11. So I'll go here. Hopefully you won't lose
track of what I'm doing. So this is 9 plus or
minus 11, over minus 20. And so if we said 9 plus 11
over minus 20, that is 9 plus 11 is 20, so this
is 20 over minus 20. Which equals negative 1. So that's one root. That's 9 plus -- because
this is plus or minus. And the other root would be 9
minus 11 over negative 20. Which equals minus
2 over minus 20. Which equals 1 over 10. So that's the other root. So if we were to graph this
equation, we would see that it actually intersects the x axis. Or f of x equals 0 at the
point x equals negative 1 and x equals 1/10. I'm going to do a lot more
examples in part 2, because I think, if anything, I might
have just confused you with this one. So, I'll see you in the
part 2 of using the quadratic equation.