I found this question in my math book:
Instability. For small |a| the equation (x - k)^2 = a has nearly a double root.
Why do these roots show instability?
I read and belive I understood the part in my book about stability and instability but it said nothing about equations, only algorithms.
I guess I don't understand the question and probably understand stability.
So please explain so that I can learn this properly
Ignore the $k$, that's basically irrelevant. The point is really that if $a,b$ are small positive numbers, then $|\sqrt{a}-\sqrt{b}|$ is much larger than $|a-b|$. You can see this with a linear approximation: $\sqrt{b} \approx \sqrt{a} + \frac{b-a}{2\sqrt{a}}.$ So $|\sqrt{a}-\sqrt{b}| \approx \frac{|a-b|}{2 \sqrt{a}}$. Thus the error in the input basically gets divided by $2 \sqrt{a}$ in the output. This is a small number, so the error in the output is much larger than the error in the input. This is what we mean by numerical instability.