>> Instructor: On this video we
solve the following quadratic
equation by completing
the square.
And for those who also
know the quadratic formula,
we then solve it
using that as well.
All right, here's
another problem
on completing the square,
where the coefficient is not 1.
So how do we begin this?
We would like the
coefficient of x to be 1.
So since the coefficient is 3,
we need to divide both
sides of the equation by 3.
Whatever the coefficient
of x squared is.
So, remember, that means
you need to divide all terms
on both sides of the
equation by that coefficient.
So this gives me x squared.
Now, 5 over 3, you have to
simply write that as 5 over 3,
five-thirds x. Now I could
write plus seven-thirds.
But what I'm going to do is
put that on the other side
of the equation, by subtracting
seven-thirds from both sides,
so I'll have a negative
seven-thirds
on the right-hand side.
Now notice I've put a
plus on both of them.
I'm going to do that to
remember we're going to have
to add something to both sides,
so that we can complete
the square
on the left side
of the equation.
All right.
Now instead of doing this all
in our head, I'd like to go down
and say, well, what would
go in this parentheses?
This would be a perfect
square of what?
Well, to get the x squared,
the first one will have
to be an x. Minus sign here
means there's a minus sign here.
And we need half of the
coefficient of x. All right.
So if you can't do that in your
head, how would you get half
of the coefficient of x. You
take half of five-thirds.
And nothing cancels,
so that is five-sixths.
So five-sixths is half
the coefficient of x.
So when I square this,
x minus 6, squared,
I will get the x squared
minus five-thirds x,
and the last term will be
whatever five-sixths squared is.
So you square the numerator,
square the denominator.
That'll be twenty-five
thirty-sixths.
Whatever you add to the
left side of the equation,
you need to add to the
right side of the equation.
Okay. So the left
side looks good.
Now we need to get a common
denominator over here,
on the right-hand side.
Okay, I'm going to move this
over just a little bit --
it doesn't want to move over.
Okay. So I'm going to
get a common denominator.
So I've got 3 in the
denominator, and 36.
I need to get a denominator
of 36.
That's the least
common denominator.
So negative seven-thirds.
Let's see, I'm going to
multiply that by 12 over 12 --
and you can do this
on scratch paper --
plus your twenty-five
thirty-sixths.
If all of you see, we now have
a common denominator of 36.
So this gives me x minus
five-sixths, squared, equals --
all right, what does
that give me?
Well, I've got negative
7 times 12.
That's going to be negative
84, plus 25, all over 36.
I'm just showing
all the steps here.
So we have x minus
five-sixths, squared, equals --
right, we're still
going to simplify.
This will give me a
negative number here.
Let's see, I think that's what?
Negative 59?
Over 36. So I've got x
minus five-sixths, squared,
equals negative 59
thirty-sixths.
So now we want to take the
square root of both sides.
Now remember, when you take
the square root of both sides,
you're going to have to put
a plus-or-minus in front
of this number on the right.
So on the left side, when
I take the square root,
I could just write the
x minus five-sixths.
But on the right side, I
have to put plus-or-minus
in front of the fraction.
So what will that be?
Well, the numerator's
the square root of 59.
Remember what that means?
That means a square root of 59i.
And the square root of
the denominator will be 6.
Well, the cool thing is at least
I have the same denominator
on both sides.
Almost done.
So this gives me two
separate equations.
But before splitting it up
in two separate equations,
I'm just going to add
five-sixths to both sides.
So I get x equals five-sixths,
plus-or-minus the square
root of 59i, over 6.
All right.
So this gives you two solutions.
We have 5 plus the
square root of 59i.
Now make sure you don't put
the i under the square root.
Some people put the i in front
of the square root,
which is also fine.
All over 6.
And then we have 5 minus
the square root of --
the square root of
59i, all over 6.
Now since these are
complex numbers,
sometimes they're written
in the form a plus bi,
so you could also write that
as five-sixths plus square root
of 59, over 6i, and
five-sixths minus square root
of 59, over 6i.
All right.
So there we go.
So our original problem was --
let's see, here it
is down here --
3x squared minus 5x
plus 7 equals zero.
And either way of
these is a correct way
of writing the solution,
so we did that by
completing the square.
Now for those of you who already
know the quadratic formula --
this is in other videos -- let's
see if we get the same answer,
if we use the quadratic formula.
So if you use the
quadratic formula,
you take the coefficients of
x squared, x and the constant,
if it's in this form, ax squared
plus bx plus c. So here a is 3,
b is negative 5, and c is 7.
And I'm going to do
b squared minus 4ac.
I like to figure that
out first, personally.
So b squared will be
a negative 5 squared.
Right? So negative 5
time negative 5 is 25.
Minus 4 times ac.
That means a times c.
Well, 3 times 7 is 21.
So that gives me 25 minus
84, which is negative 59.
So remember, b squared minus
4ac is what goes underneath the
square root in the
quadratic formula.
So the quadratic formula says
the answer is the opposite
of b. Okay.
Since b is negative
5, that'll just be 5.
Don't think I'm going to have --
to have such quite
a big line here.
So that'll be a positive
5, plus-or-minus.
All right.
Now the square root of whatever
I just found b squared minus 4ac
to be.
So it's going to
be the square root
of negative 59, all over 2a.
And since a is 3, 2a is 6.
And of course, we don't leave a
negative underneath the square
root, so that would be 5
plus-or-minus square root
of 59i, over 6, which
could also be written
as the way I have it right here.
Okay. So we get the same
answer, if you know the formula
for the quadratic formula.
If you don't know it, then
go on to my videos now
on how do you complete
the square
to get the quadratic formula.
And then look at my videos on
using the quadratic formula.
