Question

For what value of $a$ do the roots of the equation $2x^2+6x+a=0$ satisfy the condition $\frac{x_1}{x_2}+\frac{x_2}{x_1}<2$?

Answer 1

$2x^2+6x+a=0$

Sum of roots$=x_1+x_2=-3$

Product of roots$=x_1{x}_2=\frac{a}{2}$

$\frac{x_1}{x_2}+\frac{x_2}{x_1}<2$

$\frac{x_1^2+x_2^2}{x_1{x}_2}<2$

$\frac{(x_1+x_2{)}^2-2x_1{x}_2}{x_1{x}_2}<2$

$\frac{9-a}{\frac{a}{2}}<2$

$\frac{2(9-a)}{a}-2<0$

$\frac{9-a-a}{a}<0$

$\frac{2a-9}{a}>0$

$aϵ(-\infty ,0)\cup (\frac{9}{2},\infty )$