Question

Find all the values of the parameter $a$ for which both roots of the quadratic equation $x^2-ax+2=0$ belong to the interval $(0,3)$.

• A. $2\sqrt{2}<a<\frac{11}{3}$
• B. $\sqrt{2}\le a<\frac{11}{9}$
• C. $\sqrt{2}\le a<3$
• D. $2\sqrt{2}\le a<\frac{1}{3}$

Answer 1

Answer: (A)

$2\sqrt{2}<a<\frac{11}{3}$

Given equation is
$x^2-ax+2=0$
For roots to lie in the interval $(0,3)$ following conditions should hold:
1) $D\ge 0$
$\Rightarrow a^2-8\ge 0$
$\Rightarrow (a+2\sqrt{2})(a-2\sqrt{2})\ge 0$
$\Rightarrow a\in (-\infty ,-2\sqrt{2})\cup (2\sqrt{2},\infty )$ ....(1)
2) $0<-\frac{b}{2a}<3$
$\Rightarrow 0<\frac{a}{2}<3$
$\Rightarrow a>0$ and $a<6$ ....(2)
3)$a\cdot f\left(0\right)>0$
which is true for all $a>0$
4) $a\cdot f\left(3\right)>0$
$9-3a+2>0$
$a<\frac{11}{3}$ ....(3)
From equations (1),(2) and (3), we get
$a\in (2\sqrt{2},\frac{11}{3})$.