



So now we
have a step by step process to find X, 


but wow!
it is really long and involved. 


It is a
lot to remember! 


Couldn't
we maybe just have a formula 


to solve
these things all in one shot? 


Huh? 


Please? 





SURE 





For the
last two pages, we've been solving stuff like: 





3X^{2}
+ 5X + 2 = 0



and 


X^{2}
 4X + 3 = 0






Here's
what we're going to do. 


We're
going to write one of these equations as ... 





AX^{2} +
BX + C = 0 





Then
we're going to do our long process from the last page on it. 


In the
end, we will have ... 





X = some
junk with A's B's and C's in it






From then
on, when we have one of these equations 


all we
have to do is pick out the A, B, and C value 


and put
it in the formula we are about to build. 





No more
nasty factoring. 


Just
remember the formula, and you're set.



Here we
go: 





STEP
1: 


Divide
both sides of the equation by the coefficient on the X^{2}
... 











Simplify
... 











STEP
2: 


Take the
X term and determine the value of B. 


Use that
to find B^{2}. 











STEP
3: 


Put the B
^{2}
value into the equation. 











STEP
4: 


Gather up
the terms of the square, 


and write
them as the square. 











STEP
5: 


Put all
of the other baloney that's not part of the square 


on the
right side of the equation. 


Simplify
it as much as you can. 











STEP
6: 


Solve
this for X. 











And there
it is! 





The most
important formula in all of Algebra. 





It was a
lot of work getting here, 


but here
is the payoff. 





Now, if
we have something like ... 





5X^{2} 
13X + 6 = 0 





We can
say, OK: 





A =
5 B =
13 C = 6 





and put
these numbers into "THE FORMULA" 











This is
the same answer as we got before for this one, 


but this
time it was a whole lot easier. 





Our new
formula is called ... 





The Quadratic
Formula 





When we
use it, sometimes we will get two answers, 


sometimes
we will get one answer, 


and
sometimes we will not get any answers. 





If we
don't get any answers, 


it
doesn't mean that the formula didn't work. 


It means
that the problem really doesn't have any answers. 





There is
a quick way to tell how many answers you will get. 


Just look
at this part of the formula: 











If B
^{2}
 4AC works out to be a positive number, 


you will
get two answers for X. 





If B
^{2}
 4AC works out to be zero, 


you will
get one answer for X. 





If B
^{2}
 4AC works out to be a negative number, 


you will
not get any answers for X. 





Because
this piece of the formula can tell you that, 


it gets
it's own name. 


It is
called: 





The Discriminant 





Let's do
another one ... 





Example: 





4X^{2} 
20X + 25 = 0 





So: 


A =
4 B =
20 C = 25 











The
"discriminant" part of the formula was equal to zero, 


so we
only get one answer. 





Example: 





6X^{2} =
 2X + 4 





Whoa!
Time out! 


Before we
do anything, we need to move all of the values 


over to
the left side of the equation so we have: 





AX^{2} +
BX + C = 0 











So: 





A =
6 B =
2 C = 4 











copyright 2005 Bruce Kirkpatrick 
