We want to solve the quadratic equation
five x squared times
40x plus 207 equals zero
using the quadratic formula.
However, on this example we're only asked
to find the approximate solutions
where we write the answers
in approximate form
rounded to three decimal places.
Which means you don't have
to simplify the square root,
we can convert to a decimal
and round to three decimal places.
Let's go ahead and set
this up on the next slide.
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 the x squared term,
so a is five.
B is coefficient of the x term,
so b is 40.
And c is the constant term,
and therefore c equals 207.
Though we should be careful
about the signs here.
If we had subtraction
we would have a negative coefficient.
And now I'll perform substitution
into the quadratic formula.
So negative b would be negative 40.
And then we have plus or minus
the square root
of b squared.
That would be 40 squared
minus four times a,
a is five,
times c, c is 207,
divided by two times a
which would be two times five.
Now let's begin simplifying.
We have negative 40
plus or minus the square root
of this expression here,
which is called the discriminant,
I'll divide it by 10.
Let's divide with the
discriminant on the calculator.
So we have 40 squared
minus four
times five
times 207,
which equals negative 2,540.
So notice how the
discriminant is negative,
so we should recognize it will
have two complex solutions.
So we have negative 40
and then plus or minus the square root of.
We know we can write negative 2,540
as 2,540
times negative one
divided by ten.
And we know the square of negative one
is equal to i.
So let's go ahead and write this as
negative 40 plus or minus
the square root of 2,540i
divided by 10.
Now remember, we're only asked
to get the decimal approximation
of three decimal places.
So let's go ahead and write this
in the form a plus or minus b i
and then we'll get our
decimal approximations.
So we have x equals,
we'd have negative 40 divided by 10
plus or minus the square root of 2,540
divided by 10i.
Notice how the real part simplifies nicely
to negative four.
And now for the imaginary number
we'll get a decimal approximation
of three decimal places.
So we'll have plus or minus
and now we'll find this quotient
of three decimal places.
So we would have
the square root of 2,540,
right arrow,
divided by 10.
And notice here if we round
to three decimal places,
the eight tells us to round up,
so it's going to round to 5.040.
And of course times i.
Now I want to go back to the
calculator one more time.
This calculator does have complex mode.
So if we press the mode key,
notice how a plus b i a is highlighted,
which means if we go
back to the home screen
we could have entered the square root
of negative 2,540,
right arrow,
divided by 10.
And notice in this form
it gives us the decimal
approximation with the i.
Now the homework does ask us
to enter these in separately.
So our first solution x of one
is going to equal negative four
and then minus 5.040i.
And our second solution x of two
will be equal to negative
four plus 5.040i.
So going back to the previous slide,
we now have our two approximate solutions.
I hope you found this helpful.
