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Welcome to the presentation on
using the quadratic equation.
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So the quadratic equation,
it sounds like something
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very complicated.
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And when you actually first see
the quadratic equation, you'll
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say, well, not only does it
sound like something
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complicated, but it is
something complicated.
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But hopefully you'll see,
over the course of this
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presentation, that it's
actually not hard to use.
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And in a future presentation
I'll actually show you
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how it was derived.
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So, in general, you've already
learned how to factor a
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second degree equation.
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You've learned that if I
had, say, x squared minus
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x, minus 6, equals 0.
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If I had this equation. x
squared minus x minus x equals
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zero, that you could factor
that as x minus 3 and
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x plus 2 equals 0.
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Which either means that
x minus 3 equals 0 or
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x plus 2 equals 0.
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So x minus 3 equals 0
or x plus 2 equals 0.
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So, x equals 3 or negative 2.
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And, a graphical representation
of this would be, if I had the
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function f of x is equal to
x squared minus x minus 6.
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So this axis is
the f of x axis.
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You might be more familiar with
the y axis, and for the purpose
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of this type of problem,
it doesn't matter.
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And this is the x axis.
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And if I were to graph this
equation, x squared minus x,
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minus 6, it would look
something like this.
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A bit like -- this is f
of x equals minus 6.
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And the graph will kind of
do something like this.
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00:01:57,15 --> 00:02:00,03
Go up, it will keep going
up in that direction.
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And know it goes through minus
6, because when x equals 0,
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f of x is equal to minus 6.
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So I know it goes
through this point.
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And I know that when f of x is
equal to 0, so f of x is equal
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to 0 along the x axis, right?
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Because this is 1.
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This is 0.
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This is negative 1.
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So this is where f of x
is equal to 0, along
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this x axis, right?
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And we know it equals 0 at the
points x is equal to 3 and
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x is equal to minus 2.
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That's actually what
we solved here.
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Maybe when we were doing the
factoring problems we didn't
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realize graphically
what we were doing.
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But if we said that f of x is
equal to this function, we're
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setting that equal to 0.
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So we're saying this
function, when does
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this function equal 0?
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When is it equal to 0?
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Well, it's equal to 0 at
these points, right?
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Because this is where
f of x is equal to 0.
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And then what we were doing
when we solved this by
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factoring is, we figured out,
the x values that made f of x
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equal to 0, which is
these two points.
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And, just a little terminology,
these are also called
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the zeroes, or the
roots, of f of x.
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00:03:12,47 --> 00:03:14,81
Let's review that a little bit.
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So, if I had something like f
of x is equal to x squared plus
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4x plus 4, and I asked you,
where are the zeroes, or
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the roots, of f of x.
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That's the same thing as
saying, where does f of x
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interject intersect the x axis?
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And it intersects the
x axis when f of x is
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equal to 0, right?
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If you think about the
graph I had just drawn.
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So, let's say if f of x is
equal to 0, then we could
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just say, 0 is equal to x
squared plus 4x plus 4.
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And we know, we could just
factor that, that's x
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plus 2 times x plus 2.
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And we know that it's equal
to 0 at x equals minus 2.
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x equals minus 2.
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Well, that's a little
-- x equals minus 2.
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So now, we know how to find
the 0's when the the actual
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equation is easy to factor.
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But let's do a situation where
the equation is actually
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not so easy to factor.
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00:04:32,12 --> 00:04:39,75
Let's say we had f of x
is equal to minus 10x
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squared minus 9x plus 1.
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Well, when I look at this, even
if I were to divide it by 10 I
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would get some fractions here.
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And it's very hard to imagine
factoring this quadratic.
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And that's what's actually
called a quadratic equation, or
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this second degree polynomial.
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But let's set it -- So we're
trying to solve this.
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Because we want to find
out when it equals 0.
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Minus 10x squared
minus 9x plus 1.
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We want to find out what
x values make this
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equation equal to zero.
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And here we can use a tool
called a quadratic equation.
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And now I'm going to give you
one of the few things in math
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that's probably a good
idea to memorize.
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The quadratic equation says
that the roots of a quadratic
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are equal to -- and let's say
that the quadratic equation is
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a x squared plus b
x plus c equals 0.
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So, in this example,
a is minus 10.
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b is minus 9, and c is 1.
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The formula is the roots x
equals negative b plus or minus
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the square root of b squared
minus 4 times a times c,
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all of that over 2a.
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I know that looks complicated,
but the more you use it, you'll
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see it's actually not that bad.
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And this is a good
idea to memorize.
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So let's apply the quadratic
equation to this equation
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that we just wrote down.
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So, I just said -- and look,
the a is just the coefficient
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on the x term, right?
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a is the coefficient on
the x squared term.
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b is the coefficient on the x
term, and c is the constant.
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So let's apply it
tot this equation.
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What's b?
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Well, b is negative 9.
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We could see here.
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b is negative 9, a
is negative 10.
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c is 1.
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Right?
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So if b is negative 9 -- so
let's say, that's negative 9.
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Plus or minus the square
root of negative 9 squared.
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Well, that's 81.
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Minus 4 times a.
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a is minus 10.
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Minus 10 times c, which is 1.
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I know this is messy,
but hopefully you're
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understanding it.
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And all of that over 2 times a.
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Well, a is minus 10, so
2 times a is minus 20.
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So let's simplify that.
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Negative times negative
9, that's positive 9.
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Plus or minus the
square root of 81.
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We have a negative 4
times a negative 10.
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This is a minus 10.
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I know it's very messy,
I really apologize
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for that, times 1.
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So negative 4 times negative
10 is 40, positive 40.
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Positive 40.
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And then we have all of
that over negative 20.
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Well, 81 plus 40 is 121.
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So this is 9 plus or
minus the square root
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of 121 over minus 20.
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Square root of 121 is 11.
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So I'll go here.
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Hopefully you won't lose
track of what I'm doing.
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So this is 9 plus or
minus 11, over minus 20.
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And so if we said 9 plus 11
over minus 20, that is 9
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plus 11 is 20, so this
is 20 over minus 20.
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Which equals negative 1.
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So that's one root.
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That's 9 plus -- because
this is plus or minus.
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And the other root would be 9
minus 11 over negative 20.
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Which equals minus
2 over minus 20.
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Which equals 1 over 10.
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So that's the other root.
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So if we were to graph this
equation, we would see that it
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actually intersects the x axis.
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Or f of x equals 0 at the
point x equals negative
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1 and x equals 1/10.
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I'm going to do a lot more
examples in part 2, because I
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think, if anything, I might
have just confused
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you with this one.
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So, I'll see you in the
part 2 of using the
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quadratic equation.
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