Components of an Electric field due to a dipole

[forensics409]

Homework Statement

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


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

Homework Equations

E=[tex]\frac{1}{4\pi\epsilon{o}}[/tex] [tex]\frac{Q}{r^2}[/tex]
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=[tex]\frac{q}{4\pi\epsilon{o}}[/tex] [tex]\frac{((x^2+(z+[tex]\frac{d}{2}[/tex])^2)^3/2-((x^2+(z-[tex]\frac{d}{2}[/tex])^2)^3/2{((x^2+z^2)^2 +([tex]\frac{x^2d^2}{2}[/tex]-[tex]\frac{z^2d^2}{2}[/tex]+[tex]\frac{d^4}{16})^(3/2)[/tex]}}

 

[Nate810]

[/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.