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Conjugate (square roots)

Change of the sign of a square root

Summary

Change of the sign of a square root

conjugation by changing the sign of a square root

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

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

Properties

As (a + b \sqrt d)(a - b \sqrt d) = a^2 - b^2 d and (a + b \sqrt d) + (a - b \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: \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}. Hence: \frac{1}{a + b \sqrt d} = \frac{a - b \sqrt d}{a^2 - db^2}.

A corollary property is that the subtraction: :(a+b\sqrt d) - (a-b\sqrt d)= 2b\sqrt d, leaves only a term containing the root.

References

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

elementary-algebra

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