COMPLETING THE SQUARE
OBJECTIVE FOR THE DAY
The square root of -1
So, as it turns out, yes.
But, we honestly don’t know what it is, only that it exists, so we deem this number imaginary, and note it as the number i.
So, looking at our last example, we have:
Again, we know the square root of 25 is 5, -5:
And now we know that the square root of -1 is i, so:
Finally, our answer here is:
That’s how we solve it
So that’s it.
We make sure to solve the problem the exact same way as we would if the problem was positive.
And instead, we add the letter i to our answer.
This leads us to our actual lesson today which…..
What is a complex number?
A complex number is actually a combination of two numbers.
The real number, and the imaginary number.
The real number is any number that we already know about, or in other words, it’s a number that doesn’t have i attached to it.
The imaginary number is the number that has i attached to it.
So, for example:
This is actually considered a number in mathematics.
Looking at this, we can see that is has a real part:
And it has an imaginary part to it as well:
But combined, we get the actual complex number: 5 + 6i.
So, how do they happen?
Well, we’ve actually already seen an example of when they happen.
They usually happen when an expression is being squared while there is still a square that needs to be rooted.
This sounds way more complicated to explain than it is just to show, so here’s the example from yesterday.
EXAMPLE
Let’s say we have:
Again, we see the expression is squared, so let’s take the root.
So, since taking the square root is the inverse of squaring, we now have:
Now we solve:
+5 +5 +5
And x = 5 + 6i, 5 – 6i.
So, why can’t we just add them together?
Well, because there is no way to add them together.
See, it’s no different than if we were to have:
We can’t add these together in any way to get:
Nor can we add them together to get something like:
In the same way, we can’t just add the imaginary and real numbers together.
So, a complex number like:
Is as simplified as it can be.
Adding Complex numbers
So, we can’t actually add the imaginary numbers to the real numbers.
However, we can add two complex numbers together to get a single number.
Basically, we do it just like if we had some variables and numbers we were adding together.
We add the numbers together, and we add the variables together, and leave it as is.
Again, harder to explain than it is to do.
Example:
Add the following:
So, what we’re going to do in this case is we’re going to add like terms.
What that means is, we’re going to rearrange this into
Real numbers with real numbers and imaginary numbers with imaginary numbers, like so:
Now, we’re going to add them together, and get:
And there is our answer!
MULTIPLYING COMPLEX NUMBERS
Now, since we are dealing with two separate numbers, to multiply complex numbers, we need to distribute.
There are a few ways to distribute, so let’s go over those.
THE F.O.I.L. METHOD
So, let’s say we have something like:
(4 + 7i)(3 – 2i)
This is where the F.O.I.L. method can be used.
Again, remember F.O.I.L. stands for:
First
Outer
Inner
Last
It’s an organizational acronym to help you multiply what you need to without over doing it.
EXAMPLE OF THE F.O.I.L. METHOD
Following F.O.I.L. we can see that:
First:
(4 + 7i)(3 – 2i)
Outer:
(4 + 7i)(3 – 2i)
Inner:
(4 + 7i)(3 – 2i)
Last:
(4 + 7i)(3 – 2i)
So:
First = 12
Outer = -8i
Inner = 21i
Last =
But we know that:
So, our last actually becomes:
Which then becomes:
- 14
Now we put everything together
SOLVING USING THE BOX METHOD
So we know how to distribute using F.O.I.L., but are there any other easy ways to distribute?
Actually, there is; it’s called the box method.
What you do with the box method is, you put each element in an equation on the top of the box, then you put the other elements to the other equation to the side.
Then you multiply them together and come up with your answer.
(Again, we’ve reached that level in math where it’s easier to show than explain.)
EXAMPLE 1:�
Let’s use the equations that we know the answer to as an example.
So, let’s say we have:
(4 + 7i)(3 – 2i)
If we want to use the box method, here’s how we would do it:
| | |
| | |
| | |
First we make our box (always make sure to add an extra row and column to your boxes. So in this example, we have 2 elements by 2 elements, so we need a 3 by 3 box.)
Next, pick an equation and put the elements on the top of the equation.
Then put the elements of the next equation on the side of the box.
Now multiply each part individually!
4
+ 7i
3
- 2i
21i
-8i
Now add what you have together.
But we know that:
-14
NOW WE SOLVE WITH WHAT WE HAVE!
So, either way works
Like I’ve said time and time again, there’s usually at least 2 or more ways to solve a single problem.
So why are you being taught this?
Because, for some of you F.O.I.L. is the easiest way to solve these equations.
For others of you, the box method is the easiest way to solve these equations.
For the rest of you, maybe you saw these two ways and found a way that works specifically for you.
Either way, use what you think is the best way, because the reality is, that’s the way you’re going to solve these equations in the real world.
So, we know how to solve simple equations, now let’s officially define a quadratic
That’s great……so why do we care?
Because identifying each part of a quadratic equation makes it much easier to solve.
It also helps us identify what to plug in to certain equations so we can get the right answer.
So why would we want to be able to identify different parts of an equation?
Well, so far we’ve dealt with nice equations that work out well, so we’ve been getting really nice answers.
However, what happens when they don’t work out well?
Well, we can plug it into the beast (which works every single time, however it’s sort of terrible, hence the beast).
Or, we can try something a little bit easier called completing the square.
So, here’s how we do it.
Okay, so it looks like a nightmare
Example 1:
Example 2:
This works, but all of those equations were all easier two work with
Example 4:
Example 5:
THAT’S PRETTY MUCH IT
As you saw, there are times when we may get a root as an answer and that’s okay.
If you have questions about your answer though, you can always plug it back in and see if it all works out.
Of course, now you need to practice so….
Now it’s your turn!
RECAP