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CCSS.Math: , , , , , ,

I have an equation right here. It's a second degree equation. It's a quadratic. And I know its graph is
going to be a parabola. Just as a review, that means it
looks something like this or it looks something like that. Because the coefficient on the
x squared term here is positive, I know it's going to be an
upward opening parabola. And I am curious about the
vertex of this parabola. And if I have an upward
opening parabola, the vertex is going to
be the minimum point. If I had a downward
opening parabola, then the vertex would
be the maximum point. So I'm really trying
to find the x value. I don't know actually where
this does intersect the x-axis or if it does it all. But I want to find
the x value where this function takes
on a minimum value. Now, there's many
ways to find a vertex. Probably the easiest,
there's a formula for it. And we talk about where that
comes from in multiple videos, where the vertex of a
parabola or the x-coordinate of the vertex of the parabola. So the x-coordinate
of the vertex is just equal to
negative b over 2a. And the negative b, you're just
talking about the coefficient, or b is the coefficient
on the first degree term, is on the coefficient
on the x term. And a is the coefficient
on the x squared term. So this is going to be
equal to b is negative 20. So it's negative
20 over 2 times 5. Well, this is going to
be equal to positive 20 over 10, which is equal to 2. And so to find the y
value of the vertex, we just substitute
back into the equation. The y value is going
to be 5 times 2 squared minus 20 times 2 plus 15,
which is equal to let's see. This is 5 times 4, which is 20,
minus 40, which is negative 20, plus 15 is negative 5. So just like that, we're able
to figure out the coordinate. This coordinate right over here
is the point 2, negative 5. Now it's not so
satisfying just to plug and chug a formula like this. And we'll see where
this comes from when you look at the
quadratic formula. This is the first term. It's the x value that's
halfway in between the roots. So that's one way
to think about it. But another way to do
it, and this probably will be of more lasting
help for you in your life, because you might
forget this formula. It's really just try to
re-manipulate this equation so you can spot
its minimum point. And we're going to do that
by completing the square. So let me rewrite that. And what I'll do is out
of these first two terms, I'll factor out a 5, because I
want to complete a square here and I'm going to leave
this 15 out to the right, because I'm going to have
to manipulate that as well. So it is 5 times x
squared minus 4x. And then I have
this 15 out here. And I want to write this
as a perfect square. And we just have
to remind ourselves that if I have x plus
a squared, that's going to be x squared
plus 2ax plus a squared. So if I want to turn something
that looks like this, 2ax, into a perfect
square, I just have to take half of this coefficient
and square it and add it right over here in order
to make it look like that. So I'm going to do
that right over here. So if I take half of negative
4, that's negative 2. If I square it, that is
going to be positive 4. I have to be very careful here. I can't just willy nilly
add a positive 4 here. I have equality here. If they were equal
before adding the 4, then they're not going to
be equal after adding the 4. So I have to do proper
accounting here. I either have to add 4 to both
sides or I should be careful. I have to add the same
amount to both sides or subtract the
same amount again. Now, the reason why I
was careful there is I didn't just add 4 to the right
hand side of the equation. Remember, the 4 is
getting multiplied by 5. I have added 20 to the right
hand side of the equation. So if I want to make
this balance out, if I want the equality
to still be true, I either have to
now add 20 to y or I have to subtract 20 from
the right hand side. So I'll do that. I'll subtract 20 from
the right hand side. So I added 5 times 4. If you were to distribute
this, you'll see that. I could have literally, up
here, said hey, I'm adding 20 and I'm subtracting 20. This is the exact same
thing that I did over here. If you distribute the 5, it
becomes 5x squared minus 20x plus 20 plus 15 minus 20. Exactly what's up here. The whole point of
this is that now I can write this in
an interesting way. I could write this as y is equal
to 5 times x minus 2 squared, and then 15 minus 20 is minus 5. So the whole point of this is
now to be able to inspect this. When does this equation
hit a minimum value? Well, we know that this
term right over here is always going to
be non-negative. Or we could say
it's always going to be greater than
or equal to 0. This whole thing is going
to hit a minimum value when this term is equal
to 0 or when x equals 2. When x equals 2, we're going
to hit a minimum value. And when x equals
2, what happens? Well, this whole term is 0
and y is equal to negative 5. The vertex is 2, negative 5.