We want to solve the quadratic equation
three x squared plus 18
x, plus 174 equals zero
using the quadratic formula.
Let's go ahead and do
this on the next slide
where we have more room.
The first step is to identify
the values of a, b, and c,
which we'll use in the quadratic formula;
where a is the coefficient of
x squared, so a equals three;
b is the coefficient of x, so b equals 18;
and c is the constant term,
and therefore c equals 174.
And now I'll perform substitution
into the quadratic formula.
So we'll have x equals,
and then we'd have negative
b, which is negative 18
plus or minus the square root
of b squared, that would be 18 squared;
minus four times a, where a is three,
times c, where c is 174.
All of this is divided by two times a,
in this case two times three.
Now let's begin to simplify,
we'd have negative 18,
plus or minus the square root of
18 squared minus four
times three times 174.
Let's find this value, which
is called the discriminant,
on the calculator.
So we'd have 18 squared
minus four times three, times 174.
So the discriminant, or the
radicand, is negative 1,764.
Notice how because the
radicand is negative,
we know we're going to
have two complex solutions.
Our denominator is two
times three, which is six.
Now we want to simplify.
Let's write this as
negative 18 plus or minus
the square root of...
Let's write negative 1,764
as 1,764 times negative one
divided by six.
Let's check and see if
1,764 is a perfect square.
If it is, this square root
will simplify perfectly.
So we'll enter second, x
squared for the square root,
and then we have 1,764, enter.
Notice how this does simplify perfectly,
so we should be able to recognize
this will simplify to
not just 42, but to 42 i,
because we have the square
root of negative one here.
This calculator also
does have complex mode.
So another way to check this
would be to press the mode key,
make sure a plus b i is
highlighted, which it is.
Go back to the home screen,
and we should be able
to enter the square root
of negative 1,764, and it
should show us the 42 i.
So negative 1,764, enter.
And notice how it does.
So we have negative 18 plus
or minus 42 i divided by six.
And we want to be
careful simplifying here,
we cannot just simplify
the 18 and the six.
We want to separate this
into two separate fractions
and then simplify them separately.
So we have x equals
negative 18 divided by six
plus or minus 42 divided by 6 i.
Well negative 18 divided by
six equals negative three.
So we have negative three plus or minus
42 divided by six equals seven.
So we have plus or minus seven i.
Now on the homework, we
do enter these separately.
Our first solution, x sub one,
which is going to be equal to
negative three minus seven i.
And our second solution will be x sub two
equals negative three plus seven i.
So going back to the previous slide,
we have negative three minus seven i,
which would come from the
quadratic formula here
where we have minus.
And x sub two equals
negative three plus seven i,
where in the quadratic formula
we have a plus sign here.
I hope you found this helpful.
