We have equation:
[tex]4x^2 + 9x - 6 = 0[/tex]
When [tex]s[/tex] and [tex]t [/tex] are the solutions, from Vieta's formulas we know that:
[tex]s+t=-\dfrac{9}{4}[/tex]
and:
[tex]st=-\dfrac{6}{4}=-\dfrac{3}{2}[/tex]
So:
[tex]\dfrac{s}{t}+\dfrac{t}{s}=\dfrac{s^2}{ts}+\dfrac{t^2}{ts}=\dfrac{s^2+t^2}{ts}=\dfrac{s^2+t^2+2ts-2ts}{ts}=\dfrac{(s+t)^2-2ts}{ts}=\\\\\\=
\dfrac{(-\frac{9}{4})^2-2\cdot(-\frac{3}{2})}{-\frac{3}{2}}=\dfrac{\frac{81}{16}+3}{-\frac{3}{2}}=\dfrac{\frac{81}{16}+\frac{48}{16}}{-\frac{3}{2}}=\dfrac{\frac{129}{16}}{-\frac{3}{2}}=-\dfrac{129\cdot2}{16\cdot3}=\boxed{-\frac{43}{8}}[/tex]
