You get the conjugate of a binomial (two terms) by multiplying the second term by -1.
For example, the conjugate of x + y is x - y. In other words I change the sign of the second term.
The conjugate is useful because when I multiply a binomial and its conjugate, for example (x + y)(x - y), I get \(x^2 - xy + xy - y^2 = x^2 - y^2\). The middle two terms xy and -xy cancel.
You may recognize the final result \(x^2 - y^2\) as a difference of squares, in which the factored form is (x + y)(x - y).
The conjugate is especially helpful for simplifying fractions with imaginary numbers i (i is the square root of -1) because when the second term of a binomial has an i, you can multiply the binomial by the conjugate, in which the i will be squared, and \(i^2 = \sqrt{-1} ^2 = -1\), and of course -1 is more simplified and easier to deal with than i.
\(\frac{9i}{1-3i} * \frac{1+3i}{1+3i} = \frac{9i + 27i^2}{1-9i^2} = \frac{9i-27}{1+9} = \frac{9i-27}{10}\)
