Mathematics - Quadratic Equation Question with Solution | TestHub

Let $3^x=t\Rightarrow t^2-ta-a+3\le 0:t>0$

Let $f(t)=t^2-at+3-a$

Discriminate of $f(t)=0$ is $a^2-4(3-a)a$

i.e., $a^2+4a-12$.

$D\ge 0\Rightarrow a\le -6$ or $a\ge 2$.

$f(t)\le 0$. Has at least one positive solution.

If $x<0$ then at least one $t$ of $f(t)=0$ lies in.

Case I: exactly one $t\in (0,1)$ then $D\ge 0$ and $f(0)f(1)<0$ then $a\in (2,3)$

Case II: both rots lines in $(0,1)$ then $(1)D\ge 0(2)f(0)>0(3)f(1)>0$

(4) $0<\frac{a}{2}<1$

Then $a\in \varphi$

$\therefore a\in (2,3)$