The conjugate is where we change the sign in the middle of two terms like this:
We only use it 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 we multiply something by its conjugate we get squares like this:
How does that help?
It can help us move a square root from the bottom of a fraction (the denominator) to the top, or vice versa. Read Rationalizing the Denominator to find out more:
Example: Move the square root of 2 to the top:
Let's multiply both top and bottom by the conjugate (this will not change the value of the fraction):
13−√2 × 3+√23+√2 = 3+√232−(√2)2 = 3+√27
(The denominator becomes a2 − b2 which simplifies to 9−2=7)
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!