Beets, beetroot juice, inorganic nitrate, all about the same thing, different ways of consuming it. Many studies have been done looking for affects on athletic performance. They show mixed results. Some show a 2.5-4% improvement in endurance particularly. Others don't show any such thing. There are confounders aplenty. The biggest confounder is whether or not the subjects used mouthwash or a toothpaste which might damage or kill mouth bacteria. Another confounder is whether or not the subjects consumed nitrate rich vegetables in their normal diet. There are others.
https://www.pnas.org/content/109/33/13144
You have to eat a lot of beets or drink a lot of beetroot juice. The affect is dose-dependent. You're looking for a daily dosage of about 1.5g of nitrate. It's a lot cheaper to use sodium nitrate, 2g a day. Whether or not you get an athletic boost from it, it's good for your health.
https://www.ahajournals.org/doi/full...AHA.110.153536
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4008816/