Question

If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the exhaustive range of values of $a$ is

• A. $(-\frac{11}{3},\infty )$
• B. $(-\infty ,-2\sqrt{2}]\cup [2\sqrt{2},\infty )$
• C. $(-\frac{11}{3},-2\sqrt{2}]$
• D. $(-6,0)$

Answer 1

The correct option is C $(-\frac{11}{3},-2\sqrt{2}]$


Let $f\left(x\right)=x^2+ax+2$

$\to \Delta \ge 0\Rightarrow a^2-8\ge 0\Rightarrow a\in (-\infty ,-2\sqrt{2}]\cup [2\sqrt{2},\infty )\cdots \left(1\right)$

$\to 0<\frac{\text{Sum of roots}}{2}<3\Rightarrow 0<-\frac{a}{2}<3\Rightarrow a\in (-6,0)\cdots \left(2\right)$

$\to f\left(0\right)>0$ and $f\left(3\right)>0$
$9+3a+2>0\Rightarrow a\in (-\frac{11}{3},\infty )\cdots \left(3\right)$

From $\left(1\right),\left(2\right)$ and $\left(3\right)$,
$a\in (-\frac{11}{3},-2\sqrt{2}]$