How do I find $m$ for which $-2x^2 + (3m-1)x - 1=0$ would have no real roots?

Question

Find $m$ for which $-2x^2 + (3m-1)x - 1=0$ would have no real roots

I'm not certain on how to complete this question. I'm aware that the discriminant has to be less than $0$ in order for there to only be complex roots, and I can solve the question (I think) up to $$3m-1 < 2\sqrt 2$$ but I'm not sure on how to continue.

Answer 1

You obtained

$$ (3m-1)^2 < 8 $$ $$ \Leftrightarrow (3m-1) \in (-2\sqrt2, 2\sqrt2)$$ $$ \Leftrightarrow m \in \Big(\dfrac{1-2\sqrt2}{3}, \dfrac{1+2\sqrt2)}{3} \Big)$$