The Pyth. Thm. tells us the following: (hypo)^2 = (leg 1)^2 + (leg 2)^2.
Here the hypo is 75 miles. One train traveled x miles and the other x+10 miles.
applying the Pyth. Thm.,
x^2 + (x+10)^2 = 75^2
which becomes x^2 + x^2 + 20x + 400 = 5625, or
2x^2 + 20x - 5225 = 0.
Choose a method easy for you that will lead to a solution (x-value).
Using "completing the square," I obtained x= 47.8 miles and x = 57.8 miles. I
It's very important to check one's work. Suppose x = 47.8 miles. Then x+10 = 57.8 miles.
47.8^2 + 57.8^2 = 75^ must be true. Is it?
2284.84 + 3340.84 5625
5625.6 is approx equal to 5625. Close enough.
The first train traveled 47.8 miles and the second 57.8 miles.