How would I go about updating the BIOS on my Dell Inspiron 3847?

This is the latest BIOS firmware for your system  https://www.dell.com/support/home/en-us/drivers/driversdetails?driverid=5d3h8&oscode=wt64a&productcode=inspiron-3847-desktop

Scroll down and you will find installation instructions, expand it and you will see different option to apply the BIOS update.  Two main methods, you can click to run update within Windows, or update using USB via boot menu.  Easiest one is download to your desktop and double click to run it within Windows.

Updating the BIOS from Windows

Note 1: Before updating the BIOS, ensure that you suspend BitLocker encryption on a BitLocker-enabled system. If it is not enabled on your system, you can ignore this step. For information about how to disable BitLocker, see How to Enable or Disable BitLocker with TPM in Windows at support.dell.com.
Note 2: Do not turn off the power or interrupt the BIOS update process during the update.

Download and Installation
1. Click Download File, to download the file.
2. Click Save to save the file to your hard drive.
3. Browse to the location where you downloaded the file and double-click the new file.
The system restarts automatically and updates the BIOS at the system startup screen. After the BIOS update is complete, system restarts again.

Updating the BIOS from DOS (Non-Windows users)

Note 1: Before updating the BIOS, ensure that you suspend BitLocker encryption on a BitLocker-enabled system. If it is not enabled on your system, you can ignore this step. For information about how to disable BitLocker, see How to Enable or Disable BitLocker with TPM in Windows at support.dell.com.
Note 2: Do not turn off power or interrupt the BIOS update process during the update.

Installation
Note: You must provide a bootable USB drive. This executable file does not create the DOS system files.
1. Copy the downloaded file to a bootable USB drive.
2. Power on the system and then Press the F12 key.
3. Select USB Storage Device and boot to the DOS prompt.
4. Run the file by typing following command. For example:
C:\ Inspiron_3647_3847_X.exe
Where C is the drive letter of the USB device where the executable file is located and Inspiron_3647_3847_X.exe is the name of the downloaded file.
The system restarts automatically and updates the BIOS at the system startup screen. After the BIOS update is complete, system restarts again.

Updating the BIOS from DOS with UEFI boot mode enabled (Non-Windows users, Load Legacy Option Disabled)

Note 1: Before updating the BIOS, ensure that you suspend BitLocker encryption on a BitLocker-enabled system. If BitLocker is not enabled on your system, you can ignore this step.
Note 2: Do not turn off the power or interrupt the BIOS update process during the update.

Installation
1. Copy the downloaded file to a bootable DOS USB drive.
2. Power on the system and then go to BIOS Setup by pressing the F2 key.
3. Go to General > Boot Sequence > Boot > Boot List Option.
4. Change UEFI to Legacy of Boot List Option.
5. Go to Exit > Save Changes and reboot system.
6. Press F12 and then select USB Storage Device and boot to the DOS prompt.
7. Run the file by typing the file name.
The system restarts automatically and updates the BIOS at the system startup screen. After the BIOS update is complete, the system restarts again.
8. Go to BIOS Setup by pressing the F2 key and go to General > Boot Sequence > Boot > Boot List Option.
9. Change Legacy to UEFI Boot Option.
10. Go to Exit > Save Changes and reboot the system.

Another way to update in Windows is check to see if it's there in Optional Updates.  It'll be listed as firmware.  Check the box next to it.  The next time the 'Update and restart' or 'Update and power off'' options show when clicking on power, choose one and if it updates firmware, follow instructions to the letter.  It will specifically say "Do not power off..."  You may get a temporary black screen during.