Answer: x ≅ 15.95
Step-by-step explanation:
Use the pythagorean theorem for right triangles x^2 + y^2 = s^2
6^2 = 36
8^2 = 64
(x - 5)^2 requires FOIL. Unfortunately Im going to do this on paper. Its essentially First Outside Inside Last. i got x^2 - 10x + 25
Lets now add 36 and 84 and make it equal to the trinomial we just got.

Now lets subtract 120 from both sides to get:

You could complete the square, I think. But im going to use the quadratic formula.
Im going to calculate this on paper for simplicity's sake. Im getting an answer of about 
Solve by adding and then solve by subtracting to get x ≅ 15.95 and x = -5.95. We cant have a negative side length so we can disregard that negative answer.