How to solve a non-linear system of modulus equations?

Question

I have the following problem:

$$ 2x^2 + 8 \equiv 6 \;(\bmod\;13)$$ $$x \equiv 2 \;(\bmod\;15)$$

I have tried applying the Chinese remainder theorem, but could not figure out how to make it work, as this equation is quadratic. I have also tried saying $x = 2 + k \cdot 15$ and inserting this into the first equation, however that did not help either - why did I get the wrong result?.

Answer 1

Hint:

Solutions to the first equation are $x\equiv\pm5\mod13$.

Now can you apply the Chinese remainder theorem?