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Q. The number of negative integral solutions of $x^2\cdot 2^{x+1} +2 ^{|x-3|+2}$ = $x^2\cdot 2^{|x-3|+4}+2^{x-1}$ is

Complex Numbers and Quadratic Equations

Solution:

The given equation can be written as
$2^{x+1} \left[ x^2 - \frac{1}{4} \right] = 2^{|x-3|} 4 [4x^2 - 1]$
= $16\cdot 2^{|x-3|} \left[ x^2 - \frac{1}{4} \right]$
$\Rightarrow $ $2^{x-3} = 2^{|x-3|}$
[ $\because \, x^2 = \frac{1}{4}$ does not give negative integral value]
$\therefore $ $|x-3| = x - 3 $
$\therefore \, x \geq 3$
$\therefore $ given equation does not give any negative integral solution.