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Assuming that you do have a HP Compaq dx2200 Microtower PC,
The Service Manual -- which is dated January 2006 -- says that computer supports up to 2 GB of DDR-2 SDRAM

The memory sockets on the system board can be populated with up to
two industry-standard DIMMs. These memory sockets are populated
with at least one preinstalled DIMM. The sockets are labeled DIMM1
and DIMM2. To achieve the maximum memory support, you can
populate the system board with up to 2GB (2 x 1GB) of memory.

There have been several updates to the BIOS starting with 1.01 Rev. A (22 Mar 2006). The most recent is 1.10 Rev. A (18 Sep 2008).
Version 1.07 Rev. A (6 Feb 2007) -- the version referenced at the Crucial site -- is dated after the manual. Ordinarily, that would lead one to assume that there was no need to update the BIOS in order to add up to 2 GB of RAM
Furthermore, I have looked at the Release Notes for all of the versions, and none of them mentions anything about adding support for increased amounts of RAM. The closest comment is for 1.05 -- the version you already have -- which added the following feature:

Enables the selection of 256 MB of system memory to be used by the integrated video controller for the Unified Memory Architecture (UMA) if 512 MB (or more) of system memory is installed in the desktop

The enhancement added by 1.07 was support for Vista.
Some enhancements added by 1.08 were RAM-related:

Enhances the BIOS so the computer beeps when no memory is plugged in
(previously the computer would show the error only by using the LED on the
front of the computer).
- Updates ATI Surround View code in the ROM.
- Enhances the BIOS so the computer posts a Warning message when different
memory technology is installed in DIMM banks 1 and 2 (for example, if one DIMM
is using 8 IC on the board and the other DIMM is using 16 IC on the board)

Nevertheless, because Crucial is usually very reliable when it comes to issues concerning RAM (and because there were several relatively minor fixes in the various BIOS updates), I suggest that you DO update the BIOS in your computer. After you update the BIOS, directions for installing more RAM are in the Service Manual linked above at pages 1-7 to 1-9 (pdf pages 11-13)
To do so, go here --> http://h20000.www2.hp.com/bizsupport/TechSupport/SoftwareDescription.jsp?lang=en&cc=us&prodTypeId=0&prodSeriesId=1844701&swItem=ir-64576-1&prodNameId=1844702&swEnvOID=1093&swLang=13&taskId=135&mode=4&idx=0 (shortened version: http://tinyurl.com/7sw47yn)
Download sp40751.exe
The installation directions are in the tab marked Release Notes and copied below. I suggest that you close all running programs and disconnect from the Internet. Follow the installation directions exactly. If you feel uncomfortable doing this procedure, by all means take John Barnett's advice and ask a professional. I suggest going to a local independent computer repair shop, NOT a BigBoxStoreUSA or GeekSquad type of place.
INSTALLATION INSTRUCTIONS: WARNING: DO NOT TURN OFF POWER OR ATTEMPT TO REBOOT THE COMPUTER DURING THE UPDATE PROCESS!
1. Download the SoftPaq .EXE file to a directory on your hard drive.
2. Execute the downloaded file and follow the on-screen instructions.
3. Choose one of the following three options when presented:
- Create BIOS Flash Diskette.
- Create BIOS Flash DriveKey (the DriveKey must be bootable; this option creates the files needed to be able to flash the BIOS from the DriveKey).
- Flash System BIOS Now! (this option enables the BIOS to be upgraded from within the Microsoft Windows or Microsoft Vista operating system environment).
NOTE: After the files have been unpacked, the .EXE file that was downloaded in Step 1 can be deleted, if desired.

Hello, I have the following tech info for your viewing pleasure in order to find out what is the best processor upgrade I can move up to.....Property Value
Manufacturer MSI
Model 0A48
Version
Serial Number
North Bridge ATI RS400/RC400/RC410 Revision 01
South Bridge ATI SB400 Revision 01
CPU Intel(R) Pentium(R) 4 CPU 3.06GHz
Cpu Socket Socket 775 LGA
System Slots 2 PCI
Memory Summary
Maximum Capacity 1024 MBytes
Maximum Memory Module Size 1024 MBytes
Memory Slots 2
Error Correction None
******************************************
Property Value
BIOS Vendor Phoenix Technologies, LTD
Serial Number MXL6410RVM
BIOS Version MS7254 1.10
BIOS Date 08/11/2008
BIOS Size 512 KB
BIOS Starting Segment E000h
DMI Version 2.5
Characteristics - supports PCI
- supports Plug-and-Play
- supports APM
- upgradeable (Flash) BIOS
- allows BIOS shadowing
***********************************************
Property Value
HOME (Hewlett-Packard HP Compaq dx2200 MT)
Summary
Number of Logical Processors 2
Number of Physical Processors 1
CPU #1 Intel Pentium 4 524
CPU Name Intel(R) Pentium(R) 4 CPU 3.06GHz
CPU Code Name Prescott
Vendor GenuineIntel
Number of Bits 64
Instruction Set MMX, SSE, SSE2, SSE3, ET64, XD
Platform Name Socket 775 LGA
Revision G1
Technology 90 nm
Original Clock 3066 MHz
Original System Clock 133 MHz
Original Multiplier 23.1
CPU Clock 3074 MHz
System Clock 133.3 MHz
FSB 533.3 MHz
Number of Cores 1
Core #1
Speed 3066.6 MHz
Multiplier 23.0
Virtual Technology Supported No
Hyper Threading Supported Yes
Hyper Threading Enabled Yes
Cache
L1 Data Cache 16 KBytes
L1 Trace Cache 12 Kµops
L2 Cache 1024 KBytes