TITLE: Find the third rational point on the curve: $y^2 = x^3 + 8$
QUESTION [7 upvotes]: I am trying to find a third rational point on the curve $y^2 = x^3 + 8$ 
According to the my professor's solution, the idea is to find two rational points then solve for the third point.
These are the first two points:
$$(x,y) = (1, 3)~~~\text{and}~~~(x,y)=\left(-\frac{7}{4}, \frac{13}{8}\right)$$
The point $(1,3)$ is fairly straight forward and we can find it by trial and error, but we are also supposed to find the second point by trial and error. Is there a systematic way to go about this without having to try so many numbers?

REPLY [3 votes]: There is a systematic way. Draw a picture of $y^2 = x^3 + 8$ on graph paper. 
http://www.printablepaper.net/category/graph 
Per request: I am very, very old. When learning to draw graphs, I used a special kind of treeware called "PAPER," of a special type with little squares already printed on it. This special kind is called Graph Paper. If a printer is available, one may download a free pdf of graph paper, and print out a copy whenever one needs to draw a graph. 
I made one. I bought a programmable calculator in 1983. I put in $\sqrt{x^3 + 8}$ and drew the following:

Note that I bought an inexpensive flatbed scanner for my home computer, maybe about 2012, which is how you get to see the picture.