Print

Maths

Quadratic theory

Page:

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  1. Back
  2. Next

Conjugate surds

Recall that \sqrt 3 is an example of a simple surd and 2 + \sqrt 3 is an example of a compound surd.

Remember also that (\sqrt 3 )^2 is rational (namely 3=3/1) but

(2 + \sqrt 3 )^2 = 4 + 3 + 2 \times 2 \times \sqrt 3

is still not rational (that is, it can't be expressed as a fraction).

The compound surds (a + \sqrt b) and (a - \sqrt b) are called conjugate surds. They are the roots of x^2 - 2ax + a^2 - b = 0

Conjugate surds can be used to express a fraction which has a compound surd as its denominator with a rational denominator. You multiply the top and bottom of the fraction by the conjugate of the bottom line. This is useful for simplifying expressions, as shown below.

{5 \over {2 - \sqrt 3 }} = {5 \over {2 - \sqrt 3 }} \times {{2 + \sqrt 3 } \over {2 + \sqrt 3 }} = {{5(2 + \sqrt {3)} } \over {2^2 - (\sqrt 3 )^2 }} = 5(2 + \sqrt 3 )

Notice that the bottom line will always become a difference of two squares, which must be rational.

{3 \over {5 - \sqrt 2 }} = {3 \over {5 - \sqrt 2 }} \times {{5 + \sqrt 2 } \over {5 + \sqrt 2 }} = {{3(5 + \sqrt 2 )} \over {25 - 2}} = {3 \over {23}}(5 + \sqrt 2 )

Page:

  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  1. Back
  2. Next

Back to Algebra index

Links

GCSE Bitesize

GCSE Bitesize - Maths

Have you got the x-factor? Check out your factorising skills in this useful gsce bitesize activity.

Links

On the web

Links

On the web

bbc.co.uk navigation

BBC © 2012 The BBC is not responsible for the content of external sites. Read more.

This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.