Hello, my name is Susan Carroll.
I am assistant professor of Mathematics at
Georgia Highlands College, and we're going
to talk about solving quadratic equations
using the quadratic formula.
Now to solve quadratic equations, we are talking
about the generic quadratic equation – AX
squared plus BX plus C equals 0.
Now if you take this generic equation and
complete the square on that you end up with
the quadratic formula, which is X equal negative
B plus or minus square root of B squared minus
4AC over 2A.
And the terms under the radical B squared
minus 4AC, this is called the discriminant.
The discriminant lets you know how many solutions
you have and of what type of numbers they
are.
So for example, if your discriminant is a
perfect square root – say for example, square
root of 49, which is going to come up again,
okay – then that is 7, or plus or minus
7 technically, and you're going to have two
rational solutions.
Now if you have something – say like the
square root of 10, which is not a perfect
square root –then you're going to have two
irrational solutions.
Now if your discriminant ends up to be 0 – say
you'll end up with the square root of 0, which
of course is just plain 0 – then you're
going to have one rational solution, something
that can be written as a fraction.
And if you end up with a negative under the
radical – like negative 7, well that can
be simplified a little bit, as I square root
of 7.
And of course it's going to be really plus
and minus I square root of 7, so you're going
to have complex conjugate answers.
Now let's get to using this quadratic formula.
So we're going to start off with something
that actually could be factored, so we'll
get comfortable with it.
So I'm going to have X squared plus X minus
12 equals 0.
Okay, now don't forget, we need to find out
our A, which is your coefficient of X squared,
which is 1; your B, which is coefficient of
X, which is 1; and your C is your constant.
And you are going to pick up the signs in
front of these numbers.
So X equals negative B plus or minus the square
root of B squared minus 4AC, all over 2A,
is going to give you negative 1 plus or minus
the square root of 1 squared minus 4 times
1 times negative 12, all over 2 times 1.
And now that's your plug in step, which you
really shouldn't do any arithmetic on – and
now we can clean the discriminant up.
So 1 squared is 1.
We’ve got two negatives, make a positive,
4 times 1 times 12 is 48, and of course under
the radical – I mean under the fraction
bar is just a 2.
So finally, we get negative 1 plus or minus
the square root of 49 over 2 – and 49 is
a perfect square root, so you get 7 over 2
– and once the radical goes away, you pretty
much have to at this point break this up into
two problems.
Negative 1 minus 7 over 2, which is negative
8 over 2, which is negative 4 – and don't
forget to go back and do the other one.
Negative 1 plus 7 over 2, which is 2 – I
have was thinking about what the answer was
– and this is 6 over 2, which is 3.
Okay, now let's look at an example that has
a coefficient of X squared that is not just
1.
Let's do 3X squared minus 5X minus 2 equals
0.
Now this one has a couple of interesting things
going on.
Our A is 3, our coefficient of X is negative
5, and our constant is negative 2.
So once again, plugging it into our quadratic
formula – if you write this down every time
you use it, you will eventually have it memorized
– you will get negative negative 5 plus
or minus square root – now that negative
is also being squared, so put the negative
5 inside the parenthesis in the square on
the outside – minus 4 times 3 times negative
2 – and this whole thing is being held up
by the 2 times 3, so this is going to give
us positive 5 because two negatives make a
positive, and then we're going to get 25 and
that's going to be plus 24, all over 6, and
we get 49 again, underneath that radical – purely
coincidence, by the way.
All right, and of course, we know that the
square root of 49 is 7.
By now, we know that and we need to break
this up.
So X equal 5 minus 7 over 6, which is negative
2 over 6, which is negative 1/3, and X equal
5 plus 7 over 6, which is 12 over 6, which
equals 2.
Now let's look at one more problem, and this
one's going to have an issue that we need
to talk about – a couple of issues that
we need to talk about.
Let's first start off with negative 3X squared
minus 4X equals 12.
Well this is not in the form that we need
it in.
We need it set equal to 0.
So the first thing we're going to do is subtract
12 from both sides and we get that equals
0.
Now my leading coefficient is a negative.
I don't like that.
I don't like using a negative for my leading
coefficient in the quadratic formula.
So what I'm going to do is I'm going to divide
each and every one of these terms by negative
1, and it does have to be each and every one
of those terms to keep everything all balanced,
and we get 3X squared plus 4X plus 12 equals
0.
You do not have to do that step, but I find
it safe for people, to get rid of that leading
coefficient of a negative 3 instead of a positive
3.
Now this is really nice because it makes all
of our numbers positive – A is 3; B is 4;
and C is 12 – and at this point, we're ready
to go to the quadratic formula.
And once again, for the last time, writing
this thing down – and then we get negative
4 plus or minus the square root 4 squared
minus 4 times 3 times 12, all over 2 times
3, and that's going to be negative 4 plus
or minus 16 minus 144 over 6, and that's going
to give me negative 4 plus or minus the square
root of negative 128 over 6.
Now I'm going to take that down to the next
page, just so we will have enough room here.
So what we have is X equals negative 4 plus
or minus the square root of negative 128 over
6.
Okay, now negative 128 is negative 1 times
64 times 2, all over – my pen is messing
up on me again; that's a 6.
Okay, so negative 4 plus or minus 8I square
root of 2 over 6 – and you will be required
to factor out that 2 that they have in common
up in the numerator because it will cancel
with the denominator.
So 2 goes into 6 – 3 times, so we're going
to get negative 2 plus or minus 4I square
root of 2 over 3.
And if you have to break that up into separate
terms – negative 2/3 plus or minus 4/3I
square root of 2.
