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

conjugate of 3x+1 is 3x-1

We only use it 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 (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!