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

i.e., $x>0\Rightarrow t>1$

Case I: exactly one root is greater than 1

$(1)D>0(2)f(1)<0$ then $\mathrm{a}>2$

Case II: both roots greater than 1

(1) $D\ge 0(2)f(1)>0(3)\frac{a}{2}>1$

Then $a\in \varphi$

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