>> Use the quadratic formula to solve
x squared minus 4x plus 5 equals 0.
In this case, a is equal to 1, b is
equal to negative 4, and c equals 5.
Let's substitute these into the
quadratic formula and see what we get.
In this case, we have x equals
the opposite of b is positive 4,
then plus or minus the square root of b squared.
Negative 4 squared is 16.
And then we have minus 4
times a, which is 1, times c,
which is 5 all over 2 times a, which is 1.
Let's look underneath that radical.
The subtraction is going to stay a subtraction
because there's not another
negative sign to cancel out.
And 4 times 1 times 5 is going to be 20.
So underneath that radical, we have 16 minus 20.
And 16 minus 20 is negative 4.
So we're going to have negative
4 under the radical.
And 2 times 1 is just 2.
We could have actually evaluated
that in the last step.
I just didn't think about it.
Now, here we have the square root of negative 4.
We know that when we take the square root of
a negative, that becomes an imaginary number
with i. And the square root of 4 is 2.
So the square root of negative
4 is going to be 2i.
So we've got 4 plus or minus 2i over 2.
We can split this up into two
fractions so that we can reduce.
We're going to have 4 over 2 plus or minus
2 over 2 times i. And 4 divided by 2 is 2,
and 2/2 just becomes 1 and 1i is just
i. So in the end, we have x equals 2,
plus or minus i, which is our final answer.
Anytime the quantity inside the
square root ends up being a negative,
there's actually a name for that quantity.
It's called the discriminant.
Whenever that quantity is negative,
when we take the square root,
it's going to give us an imaginary number.
And this is going to lead to an
answer that is a complex number.
