



Once upon
a time a few pages ago, 


I
promised you a method to factor some equations directly 


without
all of the guessing games. 





Here it
is. 





Say you
have an equation that looks something like ... 





6X^{2}
+ 2X  4 = 0






We will
be trying to turn this into ... 





(X ± some
number)^{2} = some other number 





So how do
we do it? 





Well,
when we had (X + 4)^{2} and multiplied it out, 


we got
... 


(X + 4)^{2}
= X^{2} + 8X + 16 





In
general, using the letter "B" to stand for 


any old
number that we might have 


we get
... 


(X + B)^{2}
= X^{2} + 2BX + B^{2} 





This is
what we need to get to. 


We take
whatever we have at the start 


and make
it look like X ^{2}
+ 2BX + B ^{2} 





So
looking at 6X ^{2}
+ 2X  4 = 0, 


the
first thing we need to do is get rid of the 6 part of the 6X
^{2}. 


To do that,
we divide both sides by 6 ...






STEP
1: 


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











and
simplify ... 











STEP
2: 


Take the
X term and determine the value of B. 


Use that
to find B ^{2}. 





In this
problem, the X term is ^{1}/3X. 


Our
formula says this is equal to 2BX. 


Solve for
B ... 











So if ^{1}/6
is B then multiply it times itself 


to get B
^{2}. 








STEP
3: 


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





We can do
almost anything we want to an equation 


if we do
the same thing to both sides. 











STEP
4: 


Gather up
the terms of the square, 


and write
them as the square. 





At this
point, the first three terms on the left side 


are the
terms of our square. 


That is,
the (X + B) ^{2} = X
^{2} + 2BX + B
^{2} 


We can
substitute the square for these three terms ... 











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. 





We're
finally at the place where we have ... 





(X ± some
number)^{2} = some other number 





From here
it's easy. 


We just
peel the onion ... 











So we
have two answers ... 











If you
check back a few pages, 


you'll
see that we got the same answer there too! 





copyright 2005 Bruce Kirkpatrick 
