>> Use the quadratic formula to solve
9x squared minus 4x plus 1 equals 0.
In this case, a equals 9, b
equals negative 4, and c equals 1.
So, we can plug all of these into the
quadratic formula and see what we've got.
In this case, the opposite of b is 4, then we
have plus or minus the square root of b squared,
which is 16, minus 4 times a,
which is 9, times c, which is 1,
all over 2 times a, so that'll be 2 times 9.
If we look inside the radical, this
is going to end up being 16 minus 36,
4 times 9 times 1 is 36, and that's
all over 18, 2 times 9 is 18.
So, we've got 4 plus or minus the square root of
16 minus 36 is negative 20, again, all over 18.
Now, when we have the square root of
a negative, we know that that's going
to be an imaginary number,
so this will be 4 plus
or minus the square root
of 20 and then i over 18.
Just be careful here when you're
putting your i that it's clear
that the i isn't inside the square root.
Remember, when we have a negative in the
square root, we pull it outside as an i,
and we don't want ever to have our
math be ambiguous about whether
or not something's inside a radical.
So, just be careful with those.
I am going to continue this up here.
20 is not a perfect square, but it
does contain a perfect square factor,
so we should be able to simplify it.
20 is 4 times 5.
And so, we can take the square
root of 4, that gives us 2.
So, in the numerator, we have 4
plus or minus 2 root 5i all over 18.
And, then we can split those two terms
up into two different fractions,
so we can simplify each of them.
We have 4/18 plus or minus 2 root 5 over 18i.
4/18 reduces to 2/9 and then we have plus or
minus - we can also reduce the 2/18 to 1/9,
and so we've got that root 5 in the top,
9 in the bottom, and an i out front.
And so, our final answer here is x
equals 2/9 plus or minus root 5 over 9i.
And, that's our final answer.
