In mathematics, the conjugate of an expression of the form ${\displaystyle a+b{\sqrt {d))}$ is ${\displaystyle a-b{\sqrt {d)),}$ provided that ${\displaystyle {\sqrt {d))}$ does not appear in a and b. One says also that the two expressions are conjugate.

In particular, the two solutions of a quadratic equation are conjugate, as per the ${\displaystyle \pm }$ in the quadratic formula ${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac))}{2a))}$.

Complex conjugation is the special case where the square root is ${\displaystyle i={\sqrt {-1)),}$ the imaginary unit.

Properties

As

$${\displaystyle (a+b{\sqrt {d)))(a-b{\sqrt {d)))=a^{2}-b^{2}d}$$

$${\displaystyle (a+b{\sqrt {d)))+(a-b{\sqrt {d)))=2a,}$$

This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is:

$${\displaystyle {\frac {a+b{\sqrt {d))}{x+y{\sqrt {d))))={\frac {(a+b{\sqrt {d)))(x-y{\sqrt {d)))}{(x+y{\sqrt {d)))(x-y{\sqrt {d)))))={\frac {ax-dby+(xb-ay){\sqrt {d))}{x^{2}-y^{2}d)).}$$

$${\displaystyle {\frac {1}{a+b{\sqrt {d))))={\frac {a-b{\sqrt {d))}{a^{2}-db^{2))}.}$$

A corollary property is that the subtraction:

$${\displaystyle (a+b{\sqrt {d)))-(a-b{\sqrt {d)))=2b{\sqrt {d)),}$$

leaves only a term containing the root.