Quadratic Formula - Missing Terms
If a term is missing we will use zero in the quadratic formula.
Just as a quick reminder our formula is x equals
negative b plus or minus the square root of b squared minus 4 a c all over 2 a.
In our first example, notice that we have the equal sign here
and that our 2 x term is on the other side from our squared term.
So we want to move that over to the other side and I'll do that by subtracting 2 x from both sides,
so I now have 5 x squared minus 2 x equal to zero.
We have our a term is our squared term, so that will be five our b term is going
to be negative two it's always going to be the x and our c term will be zero.
So when I plug this into the formula I get a negative, negative two plus or minus the square root
of negative two squared minus four times our a, which is five times our c,
which is zero all over two times our a, which is five.
Now because in the second part our c was zero,
this whole thing is just goes to zero, so we can just ignore that.
So we're going to get two plus or minus the square root of negative two squared,
which would be four all over ten.
So now I have two plus or minus the square root of four,
which would be two all over ten, so two plus two which is going to give me four over ten,
which will reduce down to two-fifths, and then we have two minus two over ten which will just be zero.
So my two solutions are two-fifths and zero.
Let's take a look at example three, and this one we have everything on one side already,
so we want to identify our a, b and c, our a is always going to be the x squared term,
so that will be three our b is always the x term which means in this case there isn't one, so it will be zero
and our c is always the constant, which would be negative fifty one.
So I've got negative b, which is zero plus or minus the square root of zero squared,
minus four times our a which is three times our c which is negative fifty one,
all over two times our a which is three.
So this is going to simplify down to plus or minus the square root of our zero of course is gone,
and we have four times three, times negative fifty one, which will give us 612,
all over two times three, which would be six.
Now I don't know if I can take the square root of 612,
but I'm going to try and I get approximately 24.7, and so on I'm going to simplify this out.
612, I think is divisible by four so let's start there
and that would give me 153,
and 153 will be divisible by nine which would leave me seventeen.
So I've got so minus the square root of four times nine, times seventeen all over six.
Well, four times nine is thirty six,
and the square root of thirty-six will be six.
So we get plus or minus six, square root seventeen over six,
which means this will reduce down to plus or minus the square root of seventeen.
So now let's go on to our other examples here.
In example three, again we do not have everything all on one side, so I want to move that over.
So I get 5 x squared minus twenty three equal to zero.
So that means my a term equals five, we do not have a b term because remember that our b
is our x term and we have a c of negative twenty three.
So when I put that into our equation, we have negative b which is be zero plus or minus the square root
of b squared, minus four times our a times our c, all over two times are a.
So now this is going to simplify it down to plus or minus the square root of 460, all over ten.
460, I cannot take the square root of that but I can simplify.
It I know that it's divisible by four, so if I divide that by four I get 115
and 115 is not going to give me anything else that will simplify.
So I've got plus or minus the square root of four times 115, all over ten.
I can take the square root of four out which would be two,
so I have plus or minus two times the square root of 115, all over ten,
two and ten have a common factor of two, so that means we get plus or minus
the square root of 115 all over five.
I'll look at example four.
So again, on this one we're missing our b term, we have the a is equal to negative 2 b equals zero
and c equals thirty one.
So this looks like zero plus or minus the square root of zero squared minus
four times our a times our c, all over two times our a.
So that will simplify down to,
plus or minus the square root of 248, all over a negative four.
248 it's going to be divisible by four and it's going to give me sixty two.
Sixty two is not going to give me anything though that I can take the square root of,
so we're just going to have plus or minus square root of four times sixty two, all over negative four
square root of four is two.
So we get plus or minus two square root of sixty two, all over negative four
and they both have a common factor of two so I'll take that out.
So this could be written as a plus or minus the square root of sixty two over negative two
or plus or minus the square root of sixty two over a positive two,
and the reason why I can do that is because the negative two in the denominator will flop
the signs in the numerator.
So you still get the same x actually answer.
Whether you have a plus or minus for your two or not.
