For $2x^2-5x-12=0$, by how much does the larger root exceed the smaller one?

$x=\frac{5\pm \sqrt{25+96}}{4}=\frac{5\pm 11}{4}=4$ or $-\frac{3}{2}$. The gap is $4-(-\frac{3}{2})=\frac{11}{2}$. In general the two roots differ by $\frac{\sqrt{b^2-4ac}}{a}$, here $\frac{11}{2}$ — not $\frac{11}{4}$, which is what using $2a$ in that shortcut would give.