



Remember
when you were solving equations using the quadratic formula?












There was
a special piece of the equation called the discriminant ... 











When it
turned out to be positive, we get two values for X. 











When the
discriminant turned out to be zero, 


we get
one answer. 








But when
(b ^{2}  4ac) turned out to be a negative number, 


we were
stuck. 





A square
root means 


"what
number times itself equals the number under the radical?" 


and no
number times itself equals a negative number. 


This was
a problem we couldn't solve. 





Math
people HATE IT when that happens. 


They hate
it so much that they usually invent a new kind of math to deal with
it. 





That's
what they did here. 


They
invented a number that times itself equaled negative one. 


They
called it "i" 


They
said: 


1 × 1 = 1



That also
means that: 








Now
"i" is not a real number, it was made up. 


Maybe
they should have called it a "made up" number, 


but that
didn't sound fancy enough. 


So they
called it an "imaginary" number. 





Now that
we have "i", if you have something like: 











you can
deal with it. 





The rules
of roots say that you can do this ... 











We know
that ... 








So: 








Question: 


Is
"3i" a real number or an imaginary number? 





Hey, it's
just a name. It's not that important. 


But this
is the deal ... 


If you
have a real number times "i," you get an imaginary number. 





Enough
already, let's do one! 





X^{2}
+ X + 1 = 0






Solve for
X. 


a =
1 b = 1 c = 1









This is
all correct, 


but math
people like to see it written a little differently ... 











Take a
minute and make sure you understand how we did that. 





Whenever
you have an answer that has an "i" in it, 


math
people want it written in this special way. 


If we let
"a" and 'b" stand for any numbers we might have, 


the
special way to write this is: 


a + bi






This is
called a complex number. 





So: 








are the
places where the equation is equal to zero. 





That
means the factors of X ^{2} + X + 1 are: 











Don't
believe me? 


Multiply
it out ... 


WARNING!
It's pretty messy ... 











We got
the last term, ^{3}/4 by multiplying ... 








Make sure
you can multiply that out. 





Now
combine terms. Watch all the nasty stuff drop out ... 











Let's
try another one. 





Example: 





2X^{2}
+ 4X + 3 = 0






A =
2 B = 4 C = 3 











Now we
write this as our complex numbers ... 











are the
factors of: 


2X^{2}
+ 4X + 3






Are they
the only factors? 


Let's
multiply them out and see ... 








So we
have ... 





and we
want ... 


2X^{2}
+ 4X + 3






That
means the missing factor is a 2. 





So: 








copyright 2005 Bruce Kirkpatrick 
