The conjugate is where you change the sign in the middle of two terms like this:
It is only used in expressions with two terms (called "binomials")
|x2 - 3||x2 + 3|
|a + b||a - b|
|a - b3||a + b3|
Examples of Use
The conjugate can be very useful because ...
... when you multiply something by its conjugate you get squares like this:
How does that help?
It can help you move a square root from the bottom of a fraction to the top, or vice versa (read Rationalizing the Denominator to find out more):
Example: Move the square root of 2 to the top:
Answer: Multiply both top and bottom by the conjugate (this will not change the value of the fraction):
(Did you see how the denominator became a2-b2 ?)
Get your calculator and work out the value before and after, is it the same?
There is another example on the page Evaluating Limits (advanced topic) where I move a square root from the top to the bottom.
So try to remember this little trick, it may help you solve an equation one day!