In this example, I'm going to
use the quadratic formula to
solve the equation 3x squared
minus 5x plus 1 equals 0.
Now, before we go ahead and
assign what a, b, and c are
and actually start plugging
things into the formula, it's
absolutely critical to make sure
that we have all of the
terms on one side so that we
have a quadratic equals 0.
It has to be in that format
before we can start using the
formula to solve the equation.
Now, to use it, all we're
going to take are the
coefficients for each term.
Not the whole term, just the
number part in front.
So for this first one, we see
that a is equal to 3.
b is equal to negative 5.
And c is equal to 1.
Now again, before we assign
these, not only do we need to
make sure that everything is
on one side, but we want to
make sure that they're
in descending order.
Meaning that we have the x
squared term, then the x term,
then the constant term.
That way, the numbers match the
a, b, and c that we need
for the formula to work.
Now, it's just a matter of
plugging them in and simplify.
The quadratic formula says that
the answers for x when we
solve this will be equal to
negative b plus or minus the
square root of b squared minus
4 times a times c.
This whole thing over 2a.
And it's important to note that
it's the entire thing
over 2a, not just the radical,
and definitely not just the
stuff inside the radical.
The entire top divided by 2a.
Now, to do this, all we're
going to do is
start plugging in values.
So for our example, we have
x equal to, well,
b is negative 5.
So we want the opposite, or
negative, negative 5 plus or
minus this long square
root here.
So b squared--
again, b is negative 5.
So negative 5 squared
minus 4 times a.
a in this example is 3.
So 4 times 3 times c.
c here is 1, so times 1.
And all of that is under
our radical.
Now, we want everything we
just wrote, this entire
amount, all over 2 times
a, which again is 3.
So x will equal all of this.
This is too much
to write down.
We want to simplify it as much
as possible, but without going
to decimals.
We want to leave it exact, just
in its simplest form.
Let me move this up so that I've
got some more room here
to work with.
This is the quantity we have.
This is what we need
to work with.
So first off, let's start
dealing with what we know.
The opposite of negative 5 is
just going to be positive 5.
So let's make that
5 plus or minus.
Now, we'll start simplifying the
stuff inside the radical.
Negative 5 squared is going to
be the amount positive 25.
We want that to be minus.
We have 4 times 3 times
1, which is 12.
That whole amount over 2
times 3, which is 6.
Next up, let's simplify the
stuff inside the radical so
that we end up with 5 plus or
minus the square root of--
well, 25 minus 12 is
the amount 13.
So we end up with 13 inside
of the radical.
5 plus or minus the square
root of 13, that
whole amount over 6.
And those are our answers.
This tells us that x is equal
to these two numbers.
And though we write them
together, do realize that we
are talking about two different
values here.
We have 5 plus the square root
of 13 over 6, and 5 minus the
square root of 13,
that over 6.
So though it's easier to write
it in its condensed form, we
do actually have two
answers here.
And that's using the quadratic
formula to
solve a quadratic equation.
