What Is a Conjugate?

For $a + b$, it's $a - b$.

A conjugate is a binomial where the second term is the opposite of the second term of another binomial. In other words, if I have the expression $a + b$, the conjugate of my expression is $a - b$. If I were to add the second terms of these two expressions, I would get $0$.

Conjugates are relative to one another: two numbers can be conjugates of each other, but one number cannot simply be a conjugate in a vacuum. Conjugation also only happens between binomials: there must be exactly two terms in an equation for it to have a conjugate.

Properties of conjugates

The most significant property of conjugates is that if you multiply them together, the result will always be of the form:

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

Similarly, the roots of an equation of the form $a^2 - b^2$ will always be a pair of conjugates.

The conjugate of $5x - 13y$ is $5x + 13y$. Even if each term is far more complicated than this, you can still find the conjugate of an expression by simply flipping the sign of the second term.

(Video) What is a "Conjugate" of an Expression

by John Morse

John Morse has a good video explaining what a conjugate is.