An algebra problem by swapnil rajawat

For simplicity in writing, I'm referring to the roots as $\alpha$ and $\beta$ .

$\alpha+ \beta =-3$ and $\alpha \beta =\frac{c}{2}$ ,

So, $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^2 +\beta^2}{\alpha \beta} =\frac{(\alpha + \beta)^2 - 2\alpha \beta}{\alpha \beta} =\frac{2(9-c)}{c}=\frac{18}{c} -2$

Consider $y=\frac{18}{c} -2$ which we have to maximize for $c<0$ . It is the graph of a rectangular hyperbola in the fourth quadrant. After tracing the graph, you will find that the line $y=-2$ is an asymptote to the curve and the curve lies entirely below it. Thus, the smallest integer greater than $y$ is $\boxed{-2}$

If one root b<0 then c (constant term )must be negative because the coefficient of X is positive hence another root a must be positive So a/b is negative.... Then by inequality "if x<0 then x+1/x<2" so a/b +b /a <2