Conjugate (square roots)

Short description: Change of the sign of a square root

In mathematics, the conjugate of an expression of the form a+bd is a−bd, provided that 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 ± in the quadratic formula x=−b±b2−4ac2a.

Complex conjugation is the special case where the square root is i=−1, the imaginary unit.

## Properties

As (a+bd)(a−bd)=a2−b2d and (a+bd)+(a−bd)=2a, the sum and the product of conjugate expressions do not involve the square root anymore.

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: a+bdx+yd=(a+bd)(x−yd)(x+yd)(x−yd)=ax−dby+(xb−ay)dx2−y2d. Hence: 1a+bd=a−bda2−db2.

A corollary property is that the subtraction:

(a+bd)−(a−bd)=2bd,

leaves only a term containing the root.

## See also

* Conjugate element (field theory), the generalization to the roots of a polynomial of any degree

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