The equation $2x^2+3x+k=0$ has no real root when:

No real root means the discriminant is negative: $3^2-4(2)k=9-8k<0$, so $8k>9$ and $k>9/8$. (Forgetting that dividing by −8 reverses the inequality gives $k<9/8$, and inverting the fraction gives 8/9.) A check at $k=2$: 9 − 16 = −7 < 0, so there is indeed no real root.