>> Welcome, students.
Today, we're going to talk about the quadratic
formula, which is the capstone to this course.
It's kind of the grand finale of the course.
It's the last new topic that we're going
to cover and it is one additional way
that we can solve quadratic equations.
So, we have learned a few different
ways of solving quadratic equations.
We have used factoring, along
with the zero factor property,
we used the square root property [typing],
we learned how to complete the square -
completing the square, which we used in tandem
with the square root property, and today,
we're going to talk about the
quadratic formula [typing].
Now, you may have seen the
quadratic formula before.
I'm going to give you the
formula first and then we're going
to use the formula to go through a few examples.
In the other video below, you can actually
see, I will derive the quadratic formula.
I will show you where it's
coming from, how to develop it.
Effectively, it comes from completing
the square, but in this video,
I'm just going to give it to you and then
we are going to go through some examples.
So, here is the quadratic formula.
[ Writing ]
It says the following, the solutions to the
equation ax squared plus bx plus c equals 0,
with a not equal to 0, are given by x equals -
we've got a big fraction, the opposite of b plus
or minus the square root of b
squared minus 4ac all over 2a.
This is a formula that we can use to figure out
what the solutions are to a quadratic equation
by taking the coefficients
a, b and c and plugging them
into this formula and then evaluating it.
Now, the only reason it says a
is not equal to 0 is if a is 0,
then we have no x squareds and
it's just a linear equation.
So, obviously, in that situation, we would just
use the regular methods for solving equations,
adding and subtracting the same thing from
both sides, multiplying and dividing both sides
by the same non-zero number, etc.,
etc. This formula though is going
to work whenever we have a quadratic equation.
Now, if you want to see where this
comes from, watch the video below
where I derive the quadratic formula.
I think it's a good video to watch,
but right now, I want us to focus
on actually using this to
solve quadratic equations.
So, let's go and do an example.
Let's solve x squared minus 5x plus 2 equals 0.
Now, first we have to identify what a,
b and c are so that we can plug them in.
A is the coefficient on the x
squared, so in this case, a is 1.
It's that invisible 1.
B is the coefficient on the x
term, in this case it's negative 5.
And, c is that constant term, which is 2.
Now, we're going to plug these into the
quadratic formula and then evaluate.
So, we have x equals - and, we start off with
the opposite of b. Well, if b is negative 5,
then the opposite of negative
5 becomes positive 5.
Those negatives cancel.
Then, we have plus or minus
the square root of b squared.
Well, in this case, negative 5 squared is
25 minus 4 times 1, which is 1, times c,
which is 2, all over 2 times a, which is 1.
Once we've plugged things
in, we can go and evaluate.
So, we've got 5 plus or minus - let's
look underneath the square root.
We've got 25 minus this product,
4 times 1 times 2.
Well, 4 times 1 times 2 is 8, so that
becomes 25 minus 8, over 2 times 1 is 2.
And then, 25 minus 8 is 17.
So, we end up with a 17 inside the radical.
Now, 17 is not a perfect square and it
doesn't contain any perfect square factors,
so there's nothing we can really do here to
simplify this and none of these are going
to divide by 2, so we can
leave our answer like that.
This is fine for an answer.
You could also split it up into two fractions,
5/2 plus or minus the square root of 17 over 2.
Either one of these is acceptable as an answer.
It's generally a little bit
better to split them up,
because sometimes we will have
something that we can reduce.
And, in that case, we always want
to reduce it, but in this case,
either one of these works as an answer.
Let's take a look at another one and then,
if you want to look at more examples,
which I encourage you to do so because
all of these examples are going
to be just a little bit different, you
can look at the other videos linked below.
Let's do one more though.
Let's solve x squared plus 3x minus 10 equals 0.
Again, we need to identify a, b and c.
A is 1, b is 3, and c is negative 10.
Now, let's plug them into the formula.
We start off with the opposite of b, since
b is 3, the opposite of 3 is negative 3,
plus or minus the square root of b squared -
in this case, that's 3 squared, which is 9,
minus 4 times a, which is 1, times c, which is
negative 10, all over 2 times a - 2 times 1.
Good. Now, we can look underneath that radical.
I want to point something out here.
We're going to start off with 9, but then we are
subtracting something that's being multiplied
by a negative.
Those are going to cancel out and this
is going to turn into an addition.
And, then we've got 4 times 1
times 10, which gives us 40.
And, in the denominator, 2 times 1 is 2.
So, underneath that radical,
we now have 9 plus 40,
which we know is 49, so we
can rewrite that as 49.
So, we've got negative 3 plus or
minus the square root of 49 over 2.
Now, this is different from the last problem,
because now we have a perfect square
underneath the radical and we can evaluate that.
This becomes negative 3 plus or minus 7 over 2.
And now, we're going to want to look
at these two solutions separately,
because we can actually take negative 3 plus
7 and negative 3 minus 7 and divide those by 2
and then we'll get simpler answers.
So, we've got negative 3 plus 7 over 2
or x can be negative 3 minus 7 over 2.
Now, negative 3 plus 7 is 4, so
we've got 4 over 2, which is 2.
And, then over here, negative
3 minus 7 is negative 10
and negative 10 divided by 2 is negative 5.
So, we end up here with two solutions,
x equals 2 or x equals negative 5.
Now, one last thing before I ask you to
look at the other videos of examples,
if we look at the original equation here, this
is actually something that could be factored.
And, if we solve this by factoring, it's
going to be a lot less work than going
through and doing all these steps.
So, today, we're going to get practice just
using the quadratic formula, but after this,
when you're allowed to use
any method that you like,
I would recommend using factoring whenever
you've got a polynomial that can be factored
and saving the quadratic formula for a
situation where factoring doesn't work.
