Question

Find the number of real solutions of $(7+4\sqrt{3}{)}^{|x|-8}+(7-4\sqrt{3}{)}^{|x|-8}=14$.

Answer 1

Answer: 4

$(7+4\sqrt{3}{)}^{|x|-8}+(7-4\sqrt{3}{)}^{|x|-8}=14$
$\Rightarrow (7+4\sqrt{3}{)}^{|x|-8}+\frac{1}{(7+4\sqrt{3}{)}^{|x|-8}}=14$
$\dots \left[\Rightarrow 7-4\sqrt{3}=\frac{1}{7+4\sqrt{3}}\right]$
$(7+4\sqrt{3}{)}^{|x|-8}=t$, (say)
$\Rightarrow t+\frac{1}{t}=14$
$\Rightarrow t^2-14t+1=0$
$\Rightarrow t=\frac{14\pm \sqrt{196-4}}{2}$
$=\frac{14\pm \sqrt{192}}{2}$
$=7\pm 4\sqrt{3}$
$\Rightarrow t=7+4\sqrt{3}$
or $t=7-4\sqrt{3}=\frac{1}{7+4\sqrt{3}}$
$\Rightarrow (7+4\sqrt{3}{)}^{|x|-8}=(7+4\sqrt{3}{)}^1$
or $(7+4\sqrt{3}{)}^{|x|-8}=(7+4\sqrt{3}{)}^{-1}$
$\Rightarrow |x|-8=1$
or $|x|-8=-1$
$\Rightarrow |x|=9$
or $|x|=7$
$\Rightarrow x=9,-9$
or $x=7,-7$
$\Rightarrow$ Number of real solutions $=4$