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This video is provided as supplementary material
for courses taught at Howard Community College and in this video
I wanna show how to use the technique of completing the square
to find the vertex form of a quadratic equation.
So let's talk about the two different forms
that you might see for a quadratic equationl There's the standard form,
and that's written as y = ax-squared + bx + c,
and the a, b, and c would be
numbers, constants, and the x would be the variable.
The vertex form is going to be
y = a (x - h)-squared + k.
What we want to know is how to convert from the standard form
to the vertex form. So that's what we're going to learn.
Now before we go right into the technique,
there's one more thing I want to talk about. Over here I've got a trinomial,
x-squared + 10x +25, and that
equals the binomial squared,
(x +5)-squared I want talk about
the relationship between these two
forms of the same thing.
If I look at the trinomial,
x-squared + 10x + 25, here are some things I see:
if I to divide the coefficient
of the middle term, 10, if I divide it in half and square it,
I get the last term, I get 25. If I just divide in half,
I get the constant, the second term
of the binomial which gets squared. Let's look at another one.
I've got x-squared minus -6x +9.
The middle term is
-6x. The coefficient is -6.
When I divide that in half, I get -3,
and when I square -3 I get positive 9. That's the last term.
If I want to know what that looks like as a binomial squared,
I just take that middle term, that -6,
take the coefficient, -6,
divide it in half to get -3, and that give me
the second term of the constant of
the binomial which gets squared This works for fractions as well.
So I have x-squared + 4/3 x + 4/9,
which equals
(x + 2/3)-squared. In the trinomial,
the middle term is 4/3 x, so the coefficient is 4/3.
When I divide that in half I get 2/3.
When I square 2/3 I get 4/9,
which is the last term of the trinomial.
If I just take that middle term, take its coefficient
and divided it in half, I get the constant,
the second term, of the binomial,
(x + 2/3), which is going to be squared.
So, bearing all that in mind,
here's how we do completing the square. I've got this problem: y =
2x-squared +
12x + 3. The first thing I do is put parentheses
around the first two terms, the 2x-squared
plus 12x.
Then I want to factor what I have in parentheses
so that the first term, the x-squared term,
just has a coefficient of 1. That means I'm going to rewrite this as y =
2 times the expression
x-squared
+ 6x. We leave little room
before the parentheses, and then + 3.
Now what I want to do is work inside the parentheses.
I've got what is going to be the first two terms
of my trinomial, x-squared + 6x.
If I want this trinomial to be a perfect square,
I can create that by taking that 6x,
taking the coefficient, 6, dividing it in half
and squaring it. So I'm going to have x-squared
+ 6x + 9. That's going to be a perfect square.
Now I've just gone and added +9 inside the parentheses,
and I can't just ad numbers to an equation and have it
still balance.
So I'll have to do something else in a different part of the
right side of the equation.
I added a 9 inside the parentheses. That 9 is multiplied by 2
outside the parentheses. So, really, I added
18. What I'll do at the end of the equation
is just subtract 18. That way the equation is still balanced.
And now I'll rewrite this with a binomial squared.
I'm going to write y =
2 times... instead of x-squared + 6x + 9,
I'll have the binomial x...
and then I want one half of the
middle coefficient, the 6, which is 3. So it's (x + 3)-squared
and I'm going to take that positive 3
and -18 very the end and make that
just -15. And now I've got the vertex form
of what started out as an equation in standard form,
y = 2 (x + 3)-squared -15.
y = 2 (x + 3)-squared -15.
Sometimes you may end up with fractions that have to deal with, so here's a
slightly harder problem.
The approach is going to be exactly the same. I've got y =
3x-squared - 8x
+ 2. I'll put parentheses around the first two terms
and I wanna factor out a 3 from this,
so that the coefficient for the first term is just going to be one.
I'll have y = 3
times the expression x-squared
minus -8 over 3
x, and I'll leave some room before closing the parentheses,
plus 2. And I wanna turn
what's inside the parentheses into a perfect square.
So I take this -8/3 x,
take the coefficient, -8/3. Divide it in half.
That's going to be
-4/3. And when I square that,
I get a positive 16
over 9. So this is plus
16/9.
Now I've just added something
to this right side, so I want to subtract something as well.
I have 16/9, and I've really added 3 times 16/9.
So let's see what that is. 3 times
16/9. I can simplify this down
into 16/3. So I've added 16/3.
I want to subtract
16/3.
Now let's turn this into a binomial squared and clean up everything else.
So I'm going to have y equals
3 times x... My middle term here is
-8/3 x. The middle coefficient is -8/3.
I just want half of that. That's
-4/3. That's going to be squared.
And I've got this +2 - 16/3.
Let's see what we do with that.
Well, 2 equals
6/3, and I want 6/3
minus 16/3.
So let's see. That's going to be
-10/3. So this is minus
10/3.
So this is going to be the vertex form
of my original equation, y = 3
times the expression (x - 4/3) squared
minus 10/3. Okay, just repeat this one more time,
the basic approach is going to be
you take your first two terms, put them in parentheses,
factor them so that the
coefficient for the first term is just one.
Then we complete the square inside the parentheses
by dividing that second term in half
and squaring it. We add that amount.
We balance our equation again
and then we clean everything up and end up with the vertex form.
Okay? So practice this a few times.
It's not that hard once you get the hang of it, but it does require practice.
Take care. I'll see you next time.