Difference between revisions of "Sophie Germain Identity"
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The proof involves [[completing the square]] and then [[difference of squares]]. | The proof involves [[completing the square]] and then [[difference of squares]]. | ||
− | <div style="text-align:center;"><math> | + | <div style="text-align:center;"><math>a^4 + 4b^4 = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2</math><br /><math>= (a^2 + 2b^2)^2 - 4a^2b^2</math><br /><math>= (a^2 + 2b^2 - 2ab) (a^2 + 2b^2 + 2ab)</math></div> |
== Problems == | == Problems == |
Revision as of 18:05, 4 February 2008
The Sophie Germain Identity, credited to Marie-Sophie Germain, states that:
![$a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$](http://latex.artofproblemsolving.com/8/4/0/840d1d559b21e630ec1c813b4a7c2f37fd001e57.png)
The proof involves completing the square and then difference of squares.
![$a^4 + 4b^4 = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2$](http://latex.artofproblemsolving.com/f/e/3/fe3d5ba73cb7a2e12b6f0c1d32d76fd03fe9f75b.png)
![$= (a^2 + 2b^2)^2 - 4a^2b^2$](http://latex.artofproblemsolving.com/e/3/9/e39720a7aeb5a6243f2d814cb11a58275adf596c.png)
![$= (a^2 + 2b^2 - 2ab) (a^2 + 2b^2 + 2ab)$](http://latex.artofproblemsolving.com/8/6/5/865374908609d090260fbb68650205527d89ddfe.png)
Problems
Introductory
Intermediate
- Compute
. (1987 AIME, #14)