How to solve equations of this type My math teacher gave us some questions to practice for the midterm exam tomorrow, and I noticed some of them have the same wired pattern that I don't know if it has a name:
Q1. If: $x - \frac{1}{x} = 3$ then what is $x^2 + \frac{1}{x^2}$ equal to?
The answer for this question is 11, but I don't know how, I thought it should be 9, but my answer was wrong.
Q2. If: $\frac{x}{x + y} = 5$ then what is $\frac{y}{x + y}$ equal to?
The answer for this is -5, I also don't know how.
Q3. If: $x^4 + y^4 = 6   x^2  y^2 \land x\neq y$ then what is $\frac{x^2 + y^2}{x^2 - y^2}$ equal to?
I guess the answer for this was $\sqrt{2}$ but I'm not sure.
Any body can explain how to solve these questions, and questions of the same pattern?

Hints.
$1) x-\frac1x=3\implies x^2+\frac1{x^2}-2=9\\2)\frac y{x+y}+\frac x{x+y}=1\\3)x^4+y^4=6x^2y^2\\\implies x^4+y^4+2x^2y^2=(x^2+y^2)^2=8x^2y^2\\\implies x^4+y^4-2x^2y^2=(x^2-y^2)^2=4x^2y^2$
Divide the two and take the square root.
