section 10.3 the quadratic
formula
so in this section we're going to solve
quadratic equations
which are equations that look like
ax^2+bx+c=0
quadratic just means that the highest
exponent is
two and so we have solved these equations
by factoring
but now we're going to solve using this
formula called the quadratic formula
you just need to make sure that the
equation you start off with
is set equal to 0 and then we just
input the values of a, b and c into the
formula
a is the coefficient X squared b is the
coefficient of
X and c is your constant term so for
the first example I've worked this out
already so let's take a look at the
first example
so we have a quadratic equation
its set equal to 0 which is good and
then from here what we want to do is
just read out the values
of a, b & c that way it just makes it
easier for us to use the formula
just pay attention to the signs and so
a = 2, b = 5, and c = -3
and then we just input our values into
this formula here
so it says -5
plus or minus square root
five squared -4 times two times -3
and make sure you divide the whole thing
by two times two the next thing you want
to do is we want to simplify inside the
square root
and so we get 25 plus 24
make sure you put a plus because we have
two negatives
and then continuing in the square root
we have 25 plus 24 and that gives
the square root of 49
and then also we have multiplied two times
two which gives us four
square  of 49 is a perfect square
which comes out to be seven
what you want to do from here
is recognize that there are two answers
that's why we have the plus/minus and so
you can separate your 2 answers
one of them  is -5 plus seven over four
and then we combine we get two
over four which reduces to 1/2 and then
the second solution is -5 -7
over four which gives us -12 over 4
reduce and you get -3
so the important thing here is that you
make sure that when you get your
solution
it's completely simplified and
completely reduced
let's take a look at this second example
so this one we are going to work out
together and so we want to follow the
same steps
make sure you have a zero on one side
which we do you
and so from here me want write out what
a, b  and c are and then just simplify from
from that point on so let me show you
how to do that
so if we look at this carefully a is
equal to 9
b is -6
c is -1
we're going to input our
information into the quadratic formula
so let's see what we get we're going to
write x =
negative, negative 6
why is it like that because remember the
original formula has a negative b
right here so since the b value is
negative as well
then it'll just be minus  minus 6
plus or minus square root -6
squared -4 times nine
times -1
so c is -1
divide everything by two
times nine
we're going to simplify inside the
square root
we have positive 6
plus or minus square root 36
-6 times -6 positive 36
plus 36 divide that by
18.  continue simplifying
6 plus or minus
square root of 72 over
18 72 is not a perfect square
however we can simplify square root of
72
and so it is what you should do 
when you're solving so we don't want
leave our answer like this we want to actually
reduce
and simplify the square root and the
fraction
as as much as possible so square 72
we need to rewrite that so you have a
perfect square
so 72 is
36 times two you're going to leave two
inside the radical take the square root
of 36
next
we would like to reduce & simplify
so remember
I you can break this up into two
answers and also two fractions
let's separate these into two fractions
first
we can do that and then reduce
each one so that gives me one
third plus or minus square root of two
over three you can leave your answer like
this
or like I said you can separate your two
answers one of them is one third plus
square root of two over three and other answer
is one third minus
square root of two over three so
we have completely simplified and
reduced our
solutions
