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Conjugate

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")

Other examples:

Expression Its Conjugate
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). And I will show you how.

Note: The bottom of a fraction is called the denominator, and many square roots are irrational, so this is called "Rationalizing the Denominator"

 

Example: here is a fraction with an "irrational denominator":

How can we 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), like this:

(Did you see how the denominator became "a2-b2"?)

There is another example on the page Evaluating Limits 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!