Conjugate (square roots) explained

In mathematics, the conjugate of an expression of the form

a + b \sqrt d is a - b \sqrt d, provided that \sqrt d does not appear in and . One says also that the two expressions are conjugate.

In particular, the two solutions of a quadratic equation are conjugate, as per the

\pm in the quadratic formula x = (- b \pm \sqrt{b ^{2} -4 ac)/(}{ 2 a}) .

Complex conjugation is the special case where the square root is

i = \sqrt{ -1 }, the imaginary unit.

Properties

As$$(a\; +\; b\; \backslash sqrt\; d)(a\; -\; b\; \backslash sqrt\; d)\; =\; a^2\; -\; b^2\; d$$and$$(a\; +\; b\; \backslash sqrt\; d)\; +\; (a\; -\; b\; \backslash sqrt\; d)\; =\; 2a,$$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:$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac.$$Hence:$$\backslash frac\; =\; \backslash frac.$$

A corollary property is that the subtraction:

$$(a+b\backslash sqrt\; d)\; -\; (a-b\backslash sqrt\; d)=\; 2b\backslash sqrt\; d,$$leaves only a term containing the root.

See also

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