Welcome to a video
on solving quadratic equations by completing the square.
Let's start off by taking a look at this example.
It asked us to solve the quadratic equation
by using the square root principle.
Which means if we square root both sides
of this equation, we should be able to solve for x.
Let's give it a try.
On the left side, the square root of x minus five squared
would just be x minus five
equals number of the right side we'd have
plus or minus the square root of 28.
Now to solve for x, we can just add five to both sides.
So we'd have x equals five plus or minus square root 28.
Now the square root 28 does simplify.
So let's go ahead and do that.
28 is four times seven and four is a perfect square factor.
So we would have x equals five
plus or minus two square root seven.
So the idea behind completing a square
is we want to be able to factor the left side
of the equation so that it's a perfect square trinomial
meaning it has two of the same factors as we see here.
But in order to do that, we're going to have to take some steps
to create a perfect square trinomial.
So let's take a look at how we're going to do that.
Step one, we're going to write the equation in this form
where we move the constant term
to the right side of the equation,
and we'll also leave room, because we're going to be adding
a special number to both sides of the equation.
Next, if the leading coefficient a is not equal to one,
we do have to divide every term by a.
Third, we're going to add 1/2 times b squared
to both sides of the equation.
Remember b is going to be the coefficient of the x term.
So we'll take 1/2 of that, square it,
and add it to both sides of the equation.
Step four, factor the left side of the equation
and it should be a perfect square trinomial.
So we'll write this as a binomial squared.
Then we'll square root both sides of the equation
and solve for x.
Let's go ahead and give it a try.
First step here will be to add nine to both sides.
Next the leading coefficient is one,
so we can skip that step.
Now we'll take 1/2 of b and square it.
Since b is equal to six, we're going to take 1/2
times six and then square it.
Well, 1/2 of six would be three
squared, so that'll give us nine.
We're going to add nine to both sides.
Now there's a reason why I always like to show
this calculation and you'll see why in the next example.
So we'll add nine here and we'll add nine here.
And the idea is this should now be
a perfect square trinomial.
Let's go ahead and factor it and see.
We'll have a factor of x here and here.
The factors of positive nine that add to positive six,
that'll be plus three and plus three.
Notice we have two equal factors
which verifies this is a perfect square trinomial
and on the right side we have 18.
Let's rewrite this as the quantity
x plus three squared equals 18.
Now we'll square root both sides of the equation.
On the left side, the square root of
the quantity x plus three squared would just be x plus three
equals plus or minus square root of 18.
Don't forget your plus or minus here.
Next we'll subtract three on both sides.
So we have x equals negative three plus or minus,
now the square root of 18 is going to simplify.
18, nine times two,
nine is three times three.
So we really have the square root of
three times three times two,
which simplifies to three square root of two.
So we have negative three plus or minus
three square root of two.
Let's go ahead and take a look at another example.
This one's going to be a little more challenging.
So the first step is to subtract 10 on both sides.
Next, the leading coefficient is one,
so we'll skip step two, we'll go to step three.
We take 1/2 of b and then we square it,
so we're going to have 1/2 times negative five,
and then we're going to square that.
That'll be negative 5/2 squared,
which is equal to 25/4.
We need to add 25/4 to both sides of this equation.
Now this is supposed to be a perfect square trinomial.
But you may be asking how are you going to factor this
if you have a fraction involved?
And this is the main reason why I think it's so important
to show the calculation of 1/2 b squared.
When we go to factor this,
we know it's going to be a perfect square trinomial.
The number that goes here in this binomial factor
will come from whatever number we squared to get 25/4.
We squared negative 5/2 to get the 25/4,
so this will be minus 5/2.
So that's the main reason why I always show this step.
It makes factoring this much easier
especially when it involves a fraction.
Okay, on the right side we have negative 10 plus 25/4.
Well negative 10 over one would be negative 40 over four.
Negative 40 over four plus 25 over four
would be negative 15/4.
Now we'll square root both sides of the equation.
On the left side we have x minus 5/2 must equal,
remember this is equal to the square root of negative 15
over the square root of four.
The square root of four would be two.
The square root of negative 15 would be i square root 15.
And then don't forget the plus or minus.
So the last step would be to add 5/2 to both sides.
So our answer will be x equals 5/2
plus or minus i square root 15 all over two.
Now we do have a common denominator,
so if we wanted to, we could write this as
five plus or minus i square root 15 all over two.
It really just depends on the textbook you're reading.
Let's go ahead and try one more.
Okay, so step one we will add nine to both sides
of this equation.
The next step, our leading coefficient a is equal to three.
So we need to divide every term by three.
So now we have x squared plus 2/3 x
plus something must equal three plus something.
So now we'll take 1/2 of b and square it.
Again, we're going to show that work.
1/2 times b which is 2/3.
And then square that.
But before we square it, let's simplify it,
so we actually have 1/3 squared
which is equal to 1/9.
So we're going to add 1/9 to both sides.
Again, this is now a perfect square trinomial.
Remember the constant here will be whatever number
we squared to get the 1/9.
Looking at our work, we had a positive 1/3 squared
to give us the 1/9.
So we're going to have a plus 1/3 here in our binomial factors.
This must equal, well three over one plus 1/9,
we get a common denominator.
This'll be 27/9 plus 1/9 would be 28/9.
Now we'll square root both sides of the equation.
Here we have x plus 1/3 must equal,
now our denominator's a perfect square.
So that'll be three.
Our numerator is going to be the square of 28,
which we simplified earlier, that was two square root seven.
And of course, the plus or minus to get our two solutions.
So in our last step we will subtract 1/3 on both sides,
so we'll have x equals negative 1/3 plus or minus
two square root seven all over three.
Or if we want, we can write this as negative one
plus or minus two square root seven all over three.
That's it for completing the square.
This is a good technique for solving quadratic equations
that are not factorable.
We will also use this technique
to derive the quadratic formula in another video.
Thank you for watching.
