Components of an Electric field due to a dipole

[forensics409]

Homework Statement

The problem is: Show that the components of \vec{E} due to a dipole are given at distant points, by E_x=\frac{1}{4\pi\epsilon{o}} \frac{3pxz}{(x^2+z^2)^{5/2}} and E_z=\frac{1}{4\pi\epsilon{o}} \frac{p(2z^2-x^2)}{(x^2+z^2)^(5/2)}}


http://physweb.bgu.ac.il/COURSES/PHYSICS2_B/2009A/homework/Homework-2_files/image006.jpg

Homework Equations

E=\frac{1}{4\pi\epsilon{o}} \frac{Q}{r^2}
p=qd

The Attempt at a Solution

I have tried to break the fields of each one into vector components and add the components, however, it got really messy really quickly and after simplifying it a bit i got a ridiculous equation for just the x component, i had no clue where to go and gave up on even try to get the z component.

E_{x=\frac{q}{4\pi\epsilon{o}} \frac{((x^2+(z+\frac{d}{2})^2)^3/2-((x^2+(z-\frac{d}{2})^2)^3/2{((x^2+z^2)^2 +(\frac{x^2d^2}{2}-\frac{z^2d^2}{2}+\frac{d^4}{16})^(3/2)}}

 

[member 553083]

[/tex]I feel the answer should be far simpler than this, but i cannot see what is wrong with my attempt. Any help would be much appreciated.