Question

If all real values of $x$ obtained from the equation $4^x-(a-3)2^x+a-4=0$ are non-positive, then

• A. $a\in (4,5]$
• B. $a\in (0,4)$
• C. $a\in (4,+\infty )$
• D. none of these

Answer 1

The correct option is A $a\in (4,5]$

Let $2^x=P$, then
The equation will be $P^2-(a-3)P+a-4=0$
So, $P=\frac{(a-3)\pm \sqrt{(a-3{)}^2-4(a-4)}}{2}=2^x$\
After expanding the terms in the square root an simplifying we get:
$P=\frac{a-3+a-5}{2}$ ,$P=a-4=2^x$
$a-4=2^x$
Taking $log$ on both the sides
${\log }_2(a-4)=x$
$\Rightarrow a\in (4,5]$
Hence, option 'A' is correct.