Question

The set of all $x$ satisfying $4^{x^2+2}-\left(9\right)2^{x^2+2}+8=0$ consists of

• A. infinitely many points,
• B. finitely many points from the set of all natural numbers
• C. finitely many points from the set of all intergers
• D. exactly two integers

Answer 1

Answer: (C), (D)

The correct options are
C finitely many points from the set of all intergers
D exactly two integers

Given that $4^{x^2+2}-\left(9\right)2^{x^2+2}+8=0$
$\Rightarrow {\left(2^{x^2+2}\right)}^2-9\left(2^{x^2+2}\right)+8=0$
On substituting $t=2^{x^2+2}$ in above equation, we get
$t^2-9t+8=0$
$\Rightarrow t=1,8$
Therefore, $2^{x^2+2}=1=2^0$
$\Rightarrow x^2+2=0\Rightarrow$ no solution
and $2^{x^2+2}=8$

$\Rightarrow x^2+2=3$
$\Rightarrow x=\pm 1$
Ans: C,D