I'm pretty much a graphic-design noob. I want to take this equation from this image (I want this handwriting) and convert it into a vector image. How do I go about doing that?
You can use an online tool, like Autotracer.org, for example.
I tested it with your image, choosing either SVG or PDF, Number of Colors=2 (so only one shade of green and white), and both came out great.
You can do that with a few steps in Photoshop and Illustrator. Turn the image to be black and white, increase the contrast then invert the image so it's black writing on white background. Save the image and drop it into Illustrator. There you can use Image Trace with black and white logo setting.
I realize it's not a newbie answer, but if you have access to these two apps and want to know the steps in detail, just ask and I'll describe them.