COMPLETING THE SQUARE
WITHOUT SOLVING
OBJECTIVE FOR THE DAY
QUICK REVIEW OVER STANDARD FORM OF A CIRCLE
Alright so to recap, we found the hypotenuse of the triangle, and named it r right?
Then, we actually found out how to find the length of r with the Pythagorean theorem.
But, what if instead of leaving the hypotenuse as part of the triangle, we detach it, and spin it around?
Well, that would make something like this:
So, what does this mean?
Well, it means that if we have a circle with a center at (0,0), and a radius r, any point (x, y) is on that circle if and only if
x2 + y2 = r2.
Or, like we found before:
OKAY, BUT THAT SEEMS PRETTY SPECIFIC
Well, sorta, but not really.
Actually, if we really look at it, it helps us see how to find the radius of any circle (and therefore the equation).
But to do this, we need to be a little creative.
Deriving the standard form of a circle
So, we know we can use the distance formula to find the radius of a circle right?
But so far we’ve been looking at triangles that have one of their corners at (0,0).
But what if it doesn’t? Maybe something like this:
So, what we want to find is the length of d right?
And we can also see that the measure of AB = d, right?
So, then we know if we add:
(You know, because of the Pythagorean Theorem?)
But wait, if we look at the triangle, we can actually see that:
Becomes:
So, substituting, we can see that:
Which is the same as saying:
And solving for d, we get:
And there is the standard form of a circle
Building on what we just created, we can now say:
For a circle with center (h, k) and radius r, any point (x, y) is on that circle if and only if �
Squaring both sides of this equation will give you the standard form of an equation of a circle with center (h, k) and radius r :
(x – h)2 + (y – k)2 = r 2.
Completing the Square (the intro)
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.
Wow that’s ugly
Yeah that looked pretty intimidating, but it’s okay, we’re going to do this in baby steps.
So for today, all we are going to look at is how to set up the equation to be solved.
We’ll worry about solving some other day.
So, let’s start off by starting some problem, say:
Example 1:
So, what we were given is:
+7 +7
Now we take 6, divide it by 2, then square it. So:
Then we add it to both sides:
So, we’ve turned the quadratic into something a little easier to manage.
(Again, we’ll solve tomorrow, but let’s get this down for right now.)
Example 2:
So, what we were given is:
+64 +64
Now we take 12, divide it by 2, then square it. So:
Then we add it to both sides:
Example 3:
So, what we were given is:
+ 9 +9
Now we take -8, divide it by 2, then square it. So:
Then we add it to both sides:
So how do we find the number that goes in the equation?
Just in case you didn’t notice (which is totally understandable).
The number that goes in:
Is:
Now this seems pretty complicated, but let’s look at another example
Example 4:
So, let’s say we are given is:
+ 75 +75
Now we take -10, divide it by 2, then square it. So:
Then we add it to both sides:
As you can see, we put a (-5) in the parenthesis, but that’s because that was:
Let’s try a few more to reinforce this.
Example 5:
So, let’s say:
- 17 -17
Now we take 18, divide it by 2, then square it. So:
Then we add it to both sides:
As you can see, we put a (9) in the parenthesis, but that’s because that was:
Example 6:
Last one:
- 24 -24
Now we take -14, divide it by 2, then square it. So:
Then we add it to both sides:
As you can see, we put a (-7) in the parenthesis, but that’s because that was: