To finish off the quadratic formula let's talk about what happens when we have missing terms.
If any term is missing then it's always going to be B or C, either way you just use zero in the quadratic formula.
Now sometimes, it's actually not necessary or even desirable to use the quadratic formula.
Sometimes there are other methods that are easier, if you look at example one; that problem is actually easier
if you don't use the quadratic formula.
We have 5x squared equals 2x, some of you may think that you could divide by X but don't do that.
If you do that you will lose a solution, so never ever divide by a variable.
Instead, we are going to do what we always do; get everything on one side.
So we'll have 5X squared minus 2x equals 0, what we can do is factor out X.
So I'll write X times 5x minus 2 equals 0 then X equals 0 or 5x minus 2 equals 0.
Well 5x minus 2 equals 0 the algebra there should be very easy for you at this point.
5x equals 2, we simply finish this off by dividing by 5 and we get that X equals 2/5.
So we now have both of our answers.
X equals 0, incidentally X equals 0 will always be one of your solutions if it's C your missing, if it's the constants
that we're missing.
And other answer is 2/5.
Example two is another example of a problem that's actually a little easier if you don't use the quadratic
formula.
Here we can factor 3 out because both of those coefficients are divisible by 3 and you get 3 times
X squared minus 17 equals 0.
As I said, you never want to divide both sides of an equation by a variable.
However, you can divide both sides of an equation by a constant, that's perfectly fine.
X squared minus 17 equals 0 and instead of using the quadratic formula or factoring I'm going to show you
how to do this algebraically.
Just add 17 to both sides, we get X squared equals 17 and then we use something called the even root property
that says that X equals the square root of 17.
However, a negative number can square to become positive so that reason we actually put a little plus or
minus in front.
And as you can see as weird as it might look, this is actually quiet a bit easier then factoring or using the
quadratic formula.
However, there is still some problems where factoring or the quadratic formula are easier.
Sometimes, the quadratic formula really is the easiest course of action.
For instance, if you look at 3 and 4 here, there is nothing that I can factor out and we're not missing a constant
term so that's not going to help us.
Instead it's easier to just use the quadratic formula for these.
I'll subtract 23 from both sides here, I get that 5X squared minus 23 equals 0.
If I use the quadratic formula, A equals 5, we don't have an X term so B equals 0 and C equals negative 23.
Therefor, X will equal the negative of 0 plus or minus the square root of 0 squared minus 4 times A times C which
is 5 and negative 23 and then we simply divide that by 2 times 5.
And now we simplify, well 0 doesn't change anything so this ultimately works out to just plus or minus the square
root of negative 4 times 5 times negative 23 which comes out to 460 and then we divide that by 10.
You may not remember this because it's been a few sections but we do have the ability to simplify the
solution now, if you remember we simplify roots by using prime factorization.
So let's isolate the square root of 460 and see how that will simplify, well it turns out to simplify into 2 squared
times 5 times 23 and so we can just take the square root of 2 squared and we get 2 times the square root of 5
times 23 which just equals 2 times 115 inside a square root.
So we actually have here plus or minus 2 times the square root of 115 over 10.
And because we have that 2 in front, we now have the ability to simplify this answer.
We can cross out 2 and 10, get 5 and 1, so this comes out to the square root of 115 over 5.
We haven't really seen a lot of this so let's try that again with example 4.
With example four, A equals negative 2, B equals 0 and C equals 31.
And so we're going to do the same thing here, just the same quadratic formula.
X is going to equal the negative, again B is 0 so I'll just put 0 out here, you can probably ignore that but it's a
good idea to practice writing down the entire formula just so that you'll remember it that much faster.
Again you do need to have it memorized and for A and C we're just going to put negative 2 and 31.
Then we're going to divide this whole thing by 2 times negative 2.
And yet again we simplify we get plus or minus, we can ignore negative 0, negative 4 times negative 2 times 31 is
248 we'll divide that by negative 4 and we're gong to simplify the root again but one thing that you may find
beneficial is to take out a negative from the top and the bottom because we really don't like having a negative on
the top or the bottom.
You can multiply or divide by negative 1 as long as you do it to both the top and the bottom.
It doesn't matter, it comes out the same either way.
If you have plus or minus on top multiplying that by a negative won't really change anything because the
positive will become a negative, the negative will become positive, either way it's exactly the same as it was.
So we just get plus or minus the square root of 248 divided by 4 and as I was saying, we're now tasked with
simplifying this root.
Well, if you prime factorize it; this prime factorizes into plus or minus 2 squared, actually let's isolate the square
root of 248.
So we have the square root of 248 and this breaks down into 2 cubed times 31, 2 cubed times 31.
Well if you remember, if you don't see an index of your radical you assume it's 2.
That means that we can set aside a 2 of these twos, we'll have 2 squared times 2 to the first, I'm just splitting
that up so I can take out 2 of them, times 31.
The square root of 2 squared is 2 and inside we have the square root of 2 times 31 which is 62 and now we can
go back to our main equation here.
We have plus or minus 2 root 62 over 4 and now we can cap this off just by simplifying 2 and 4 divide.
We get 2 and 1, you can not divide inside the radical so don't do that, that's a whole different thing, anyway this
whole thing now comes out to plus or minus the square root of 62 over 2.
And now we're finally done, so remember sometimes you do have to simplify roots on the inside.
If you see any square factors of whatever your root is, you do have to simplify.
A lot of the time you get lucky and it will come out to actually be a square number but sometimes you don't
and in those cases you are going to have to try simplifying as best as you can.
