WEBVTT
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Hi and welcome.
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This is Anthony Varela.
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And today, I'm going to
introduce quadratic equations.
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Now, quadratic
equation's a big topic.
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So I'm going to introduce
a lot of stuff, but not
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in a lot of detail.
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So feel free to look up these
concepts in greater detail
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by seeking out other videos.
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So we're going to talk about
quadratic relationships.
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Then we're going to
look at different forms
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of quadratic equations.
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We'll look at what quadratic
equations are like on graphs
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and, finally, some methods for
solving quadratic equations.
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So first, let's talk about
quadratic relationships.
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Well, a quadratic is a
second-degree polynomial
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with an x squared term as
its highest degree term.
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So you're not going
to see x cubed.
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You're not going to see x to
the fourth or anything higher
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than x to the power of 2.
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That's what defines a quadratic.
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So here's an example
of a quadratic,
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x squared plus 3x minus 1.
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Now, quadratic relationships
can be recognized
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from a table of values.
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So I'm going to look at
consecutive x-values.
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Let's plug in 0, 1,
2, 3, and 4 into x.
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So when x equals 0, we
have 0 plus 0 minus 1.
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So that's a negative 1.
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When x equals 1, we
have 1 plus 3 minus 1.
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So it'd give us a positive 3.
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When x equals 2, we
have 4 plus 6 minus 1.
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So that gives us a positive 9.
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When x equals 3, we
have 9 plus 9 minus 1.
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So that gives us 17.
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And finally, when x equals 4,
we have 16 plus 12 minus 1.
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So that gives us a total of 27.
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Now we're going to be
finding the difference
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between these numbers here.
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So the difference between
negative 1 and 3 is 4.
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The difference
between 3 and 9 is 6.
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The difference
between 9 and 17 is 8.
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And maybe you're already
seeing the pattern.
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The difference between
17 and 27 is 10.
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Now, in linear relationships,
if you took this difference,
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you would get the same
value, a common difference,
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as you went down the rows.
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But in our quadratic
relationships,
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we have to find the
second difference.
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So the difference between 4 and
6 is 2, between 6 and 8 is 2,
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and between 8 and 10 is 2.
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So we have the same
second difference
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in our quadratic relationships.
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So now let's talk about
forms of quadratic equation.
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So the first form that
we use is standard form.
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And this is y equals ax
squared plus bx plus c.
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So we have an x squared term,
an x-term, and our constant, c.
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So a is the coefficient
of the x squared term.
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B is the coefficient
of the x-term.
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And our constant is c.
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And we use the
standard form if we
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would like to solve a
quadratic equation using
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the quadratic formula.
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And we'll get to
that in a minute.
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Another form is
called vertex form.
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And this is y equals a times
x minus h, quantity squared,
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plus k.
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And we use equations
in vertex form
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if we'd like to easily identify
the vertex of a parabola.
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And we'll talk about parabolas
and vertices in a minute
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as well.
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And lastly, we have
the factored form
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of a quadratic equation,
which is y equals a times x
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minus x1 times x minus x2.
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And we use equations
in factored form
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if we'd like to easily
identify x-intercepts.
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So there are our different forms
of our quadratic equation--
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standard form, vertex
form, and factored form.
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Now I'd like to talk
about parabolas,
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which are quadratic
equations graphed
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on the coordinate plane.
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So a parabola is the shape of a
quadratic equation on a graph.
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It is symmetric at the vertex.
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So we're going to
talk about symmetry.
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And we'll talk about
the vertex as well.
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So the vertex of a parabola is
the maximum or minimum point
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of a parabola.
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And it's located on
the axis of symmetry.
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So here in the graph, I
have marked the red dot.
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That's the vertex
to this parabola.
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It happens to be a minimum point
because this parabola opens
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upwards.
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It's a U-shaped parabola.
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And we see that
dotted vertical line.
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That's the axis of symmetry.
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And we can think about that
as a line of reflection.
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So notice that our
parabola is symmetrical.
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So what this means is you can
take a point on the parabola
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and reflect it across that line.
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And you'll still
be on the parabola.
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Now, I said that the
vertex represents
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a maximum or a minimum point.
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Here we see it's
a minimum point.
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This is what the vertex looks
like as a maximum point.
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So you notice here the
parabola opens downward.
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And we still have
that axis of symmetry
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that acts as a
line of reflection.
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So when do we have
parabolas that are the U
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shape, opening upwards?
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And when do we
have parabolas that
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have that upside down U
shape, opening downward?
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Well, that depends
on our variable a.
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If a is a positive number,
our parabola opens upward.
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If a is a negative number,
our parabola opens downward.
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So lastly, I'd like to
talk about solutions
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to quadratic
equations and how we
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can solve quadratic equations.
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Well, first, there are a
couple of different names
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for solutions.
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We can call them roots.
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And we can also call them zeros.
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And these represent x-intercepts
on our graph of a parabola.
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So they're x-values
that make y equal 0.
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So when we're solving
a quadratic equation,
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we have our quadratic
expression set equal to 0.
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And when we solve for
x with y equals 0,
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we've found our solution.
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So there are two common ways
to solve quadratic equations.
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We could solve by factoring.
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And we can solve by using
the quadratic formula.
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So when we're factoring, we're
taking our quadratic equation
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and we're writing it as factors.
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So one factor here is x plus 1.
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And another factor is x plus 2.
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And if you're interested,
you can FOIL this out.
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And you'll get this expression.
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But in general, you notice that
x could have coefficients here.
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And the great thing about
solving by factoring
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is that once you have
it written in factors,
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you can set each factor
equal to 0 and solve for x.
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So we're going to do that
over here on the left,
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just separating this
into two equations,
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setting each factor equal to 0.
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So I can see that x equals
negative 1 and x equals
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negative 2 are solutions
to this equation.
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Now, we could also solve by
using the quadratic formula.
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And the quadratic
formula is useful
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because you can use this
for any quadratic equation.
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If you can't factor an
equation or you don't know how,
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you can always use
the quadratic formula.
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And the quadratic formula
relies on our variables a, b,
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and c in our standard
form set equal to 0.
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So the quadratic
formula is x equals
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negative b plus or minus
the square root of b
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squared minus 4ac, all over 2a.
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So like I said,
this could be used
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for any quadratic equation.
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The trade-off is that it's
kind of messy algebraically.
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But if we plugged
in-- let's see--
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a would be 1, b would
be 3, and c would be 2--
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into our quadratic formula
and did our simplification,
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we would get that x equals
negative 3 plus or minus 1,
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all over 2.
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So what you would do then
is evaluate negative 3
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plus 1 over 2 and negative
3 minus 1, all over 2.
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And we would get
x-values of negative 2
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and negative 1, same
solutions as before.
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So let's review our introduction
to quadratic equations.
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We talked about how a quadratic
is a second-degree polynomial.
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We have a couple
of different forms
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of quadratic equations--
standard form, vertex form,
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and factored form.
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When graphed, quadratic
equations are parabolas.
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And we can have parabolas
that open upward or downward,
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depending on the value of a,
if it's positive or negative.
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And we also talked
about common ways
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to solve quadratic equations,
either by factoring or using
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the quadratic formula.
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So thanks for watching
this introduction
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to quadratic equations--
hope to see you next time.