Question

If $2^{2x+4}-17\times 2^{x+1}=-4$, then which of the following is true?

• A. x is always positive value
• B. x is always negative value
• C. x can be either a positive value or a negative value
• D. x can take only one integral value

Answer 1

Answer: (C)

Given:

$2^{2x+4}-17\times 2^{x+1}=-4$
Let, $2^{x+1}=y$

$\Rightarrow (2^{x+1}{)}^2=y^2$
$\Rightarrow 2^{2x+2}=y^2$

Now, lets take $2^{2x+4}-17\times 2^{x+1}=-4$
$\Rightarrow 2^2\left(2^{2x+2}\right)-17\times 2^{x+1}=-4$
$\Rightarrow 4y^2-17y+4=0$
$\Rightarrow 4y^2-16y-y+4=0$
$\Rightarrow 4y(y-4)-1(y-4)=0$
$\Rightarrow y=\frac{1}{4}$ or $y=4$

Case 1:

$y=\frac{1}{4}$

$\Rightarrow 2^{x+1}=\frac{1}{2^2}$

$\Rightarrow 2^{x+1}=2^{-2}$ [$\because \frac{1}{a^m}=a^{-m},a\ne 0$]

$\Rightarrow x+1=-2$

$\Rightarrow x=-2-1$

$\Rightarrow x=-3$---(1)

Case 2:

$y=4$

$\Rightarrow 2^{x+1}=4$

$\Rightarrow 2^{x+1}=2^2$

$\Rightarrow x+1=2$

$\Rightarrow x=2-1$

$\Rightarrow x=1$---(2)

From (1) and (2)

$x$ can be either a positive value or a negative value.

Hence, Option C is True.