



So now we
have a great way to solve equations that look like: 





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






What
about stuff like ... 











If you
look really close, 


you will
see that the exponents on the X terms on the left 


are
exactly the squares of the exponents on the X terms in the center. 


We know
that ... 





X^{2}
= X × X






But these
are also squares ... 











Big deal.
So what? 





OK,
here's what we do. 


We chose
a letter (usually "u") 


to stand
for the X term in the middle of the equation. 


That way,
the X term on the left is u^{2}. 


Here goes
... 





Example: 





Solve
this puppy ... 











Now
substitute "u" and "u ^{2}" into the
equation for the X's ... 





u^{2} 
6u + 5 = 0 





The
Quadratic Formula works for any letter, not just X. 


That
means we can use it here. 





Lucky us! 

















But u = ,
so now put the back
in for u and find X ... 











Let's do
another one ... 





Example: 











u = X
^{2}
so we put X ^{2} back in to find X ... 











Since
there are not any real numbers that you can square 


to get a
negative three, that one has no answer. 


So the
only answer to this one is X = 1 





These
things could get really involved. 


The trick
is to just look at the thing you replace with the u 


as a
glob. 


Then just
forget about it until you get done 


with the
rest of the problem. 





Here's a
nasty one. 





Example: 





4(2X^{2}
+ 6X)^{2} + 12(2X^{2} + 6X) + 5 = 0 





Substitute
... 


u = (2X^{2}
+ 6) so u^{2} = (2X^{2} + 6)^{2}






We have
... 








So great,
we found u. 


Now we
need to put back the thing that we substituted for 


(2X
^{2}
+ 6X) so ... 











Now move
the numbers over to the left side of the equation 


so that
we can use the Quadratic Formula ... 











So ... 











That one
was pretty messy. 


But the
deal is that no matter what kind of mess you have, 


if the
variable glob on the left is the square 


of the
variable glob in the middle you can
use this trick. 





copyright 2005 Bruce Kirkpatrick 
