Question

If x be real, then solve the following equations :
$(144{)}^{|x|}-2(12{)}^{|x|}+a=0$

Answer 1

Given ,

equation is $(144{)}^{|x|}-2(12{)}^{|x|}+a=0$

$(12{)}^{2|x|}-2(12{)}^{|x|}+a=0$

Put $t=(12{)}^{|x|}$

$\Rightarrow t^2-2t+a=0$

$\Rightarrow t=1\pm \sqrt{1-a}$ and $1-a>0$

$\Rightarrow a<1$

i.e $(12{)}^{|x|}=1\pm \sqrt{1-a}$

Since $x\in R\Rightarrow |x|\in [0,\infty )$

$\Rightarrow (12{)}^{|x|}\in [1,\infty )$

$\Rightarrow (12{)}^{|x|}=1+\sqrt{1-a}$ (since $1-\sqrt{1-a}<0$)

$|x|={\log }_{12}(1+\sqrt{1-a});a<1$

$x=\pm {\log }_{12}(1+\sqrt{1-a});a<1$