The conjugate is where we change the sign in the middle of two terms like this:

conjugate of 3x+1 is 3x-1

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 we multiply something by its conjugate we get squares like this:

(a+b)(a-b) = a^2 - b^2

How does that help?

It can help us 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:

1 / (3-square root 2)

Answer: Multiply both top and bottom by the conjugate (this will not change the value of the fraction):

conjugate example

(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!