



So what
do we do with something like:






X^{2} +
4X = 0 





X
^{2}
and X are not the same thing. 


We can't
just combine them together. 


BUT, what
we can do is something called factoring. 





What's
that? 





Well you
know that X times X equals X
^{2}. 


and also
X times 4 equals 4X so: 





X times (X
+ 4) equals X^{2} + 4X 





STOP!
TIME OUT! WHOA! HOLD ON! 





Let's go
through that one more time in slow motion ... 





OK, OK. 





If we
multiply 6 times 12, 


what do
we actually do? 











First we
multiply the 6 times the 2 


and write
that down. 





Then we
multiply the 6 times the 10 


and write
that down. 





The 12 is
like a 10 plus a 2. 











Then we
add the 12 and the 60 


and get
72 ... 











OK, that
one we know. 


Now we
have X + 4 times X 











We
multiply X times X and write that down. 


Then we
multiply the X times 4 and write that down. 


Then we
add those two together ... 











It's a
little tougher to start with X
^{2} + 4X 


and
figure out what we multiplied together to get it. 





But hey,
we're big shot algebra types now, 


we can
handle it! 





That
means we can rewrite: 





X^{2}
+ 4X = 0 as X(X + 4) = 0 





Great,
just what does THAT get us? 





Well,
it's like this, 


if you
multiply two things together and get zero 


then one
or the other of the things 


MUST be
equal to zero. 





So in X(X
+ 4) = 0 we know that either 


X = 0 or
X + 4 = 0 





For the X
= 0 part, there's nothing to do. 


X = 0 and
that's that. 





But we
need to do a little work 


to find X
when we have X + 4 = 0. 


It's an
easy one ... 











So our
two answers are X = 0 and X = 4. 


Either
one of those numbers could be put in place of X 


in the
equation ... 





X^{2} +
4X = 0 





and the
thing would be true. 





Let's
check and see ... 








Since the
terms we get at the end 


on each
side of the = ARE equal to each other 


the
answers we are testing do work. 





copyright 2005 Bruce Kirkpatrick 
