Diamond Puzzles!

Jelly, 15 11 10 bytes

Hð+,_ðH²_½

Try it online!

The following binary code works with this version of the Jelly interpreter.

0000000: 48 98 2b 2c 5f 98 48 8a 5f 90  H.+,_.H._.

Idea

This is based on the fact that

Code

Hð+,_ðH²_½    Left input: s -- Right input: p

 ð   ð        This is a link fork. We define three links, call the left and right
              link with the input as arguments, then the middle link with the
              results as arguments.

H             Left link, dyadic. Arguments: s p

H             Halve the left input.

     ðH²_½    Right link, dyadic. Arguments: s p

      H       Halve the left input.
       ²      Square the result.
        _     Hook; subtract the right input from the result.
         ½    Apply square root to the difference.

 ð+,_         Middle link, dyadic. Arguments: (results of the previous links)

  +           Compute the sum of the results.
    _         Compute the difference of the results.
   ,          Pair.

Unicorn, 4650 2982 1874 1546

[ ✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨  (    ✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨     ( ) )  2         ✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨  (  ✨✨✨✨✨✨✨   4   ✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨     ( ) )    ✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨✨ ]

Now with goats, rainbows, and sparkles!

Hopefully shorter than Java.

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Uses a custom encoding which can be applied with ApplyEncoding

Explanation

How does this work??? With the magic of unicorns (and a little code).

Unicorn is compiled into JavaScript

Each section is separated by a space, and each section represents a character in the JavaScript code.

If the section contains unicorns, the section's character is the section's length, converted to a char code (e.g. 32 unicorns would be a space)

If the section contains goats, the section's length is doubled and then converted to a char code.

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If the program's special chars don't show, here's a picture:

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This is non competing because Unicorn was made after this challenge was posted.

JavaScript ES6, 45 39 37^* bytes

(q,p)=>[x=p/2+Math.sqrt(p*p/4-q),p-x]

^{* Thanks to Dennis!}