solving this radical equation; finding the value of q?

Question

So, here's my question:$\sqrt{\frac{q^{2}}2 + 11} = q - 1$

I solved it up to this point: $ q^{2} - 4q - 20 = 0$

The answer is $q = 2 + 2\sqrt{6}$

I'm missing a step... what is it? How should I get this final answer?

Answer 1

$$q^2-4q-20=0$$ $$q^2-4q=20$$ $$(q-2)^2=20+4$$ $$(q-2)^2=24$$ $$q-2=\pm\sqrt{24}$$ $$q=2\pm\sqrt{24}$$ One of them works, the other one is extraneous

Answer 2

Because you need to write $q\geq1$ before.

Now, we need to solve $$\frac{q^2}{2}+11=(q-1)^2$$ or $$q^2-4q-20=0,$$ which gives $$q=2+\sqrt{24}$$ or $q=2-\sqrt{24}$, which is impossible.