Question

Let s and t be the solutions of the quadratic 4x^2 + 9x - 6 = 0. Find

Answer 1

We have equation:

$4x^2 + 9x - 6 = 0$

When $s$ and $t$ are the solutions, from Vieta's formulas we know that:

$s+t=-\dfrac{9}{4}$

and:

$st=-\dfrac{6}{4}=-\dfrac{3}{2}$

So:

$\dfrac{s}{t}+\dfrac{t}{s}=\dfrac{s^2}{ts}+\dfrac{t^2}{ts}=\dfrac{s^2+t^2}{ts}=\dfrac{s^2+t^2+2ts-2ts}{ts}=\dfrac{(s+t)^2-2ts}{ts}=\\\\\\= \dfrac{(-\frac{9}{4})^2-2\cdot(-\frac{3}{2})}{-\frac{3}{2}}=\dfrac{\frac{81}{16}+3}{-\frac{3}{2}}=\dfrac{\frac{81}{16}+\frac{48}{16}}{-\frac{3}{2}}=\dfrac{\frac{129}{16}}{-\frac{3}{2}}=-\dfrac{129\cdot2}{16\cdot3}=\boxed{-\frac{43}{8}}$