Question

Find the value of x for the expression 4(4^x-2^x)+1=0

Answer 1

x = -1

Explanation:

`4(4^x-2^x)+1=0\\\\4\bigg((2^2)^x-2^x\bigg)+1=0\qquad\text{use}\ (a^n)^m=a^(nm)\\\\4(2^(2x)-2^x)+1=0\\\\4\bigg((2^x)^2-2^x\bigg)+1=0\qquad\text{substitute}\ 2^x=t>0\\\\4(t^2-t)+1=0\qquad\text{use the distributive property}\\\\4t^2-4t+1=0\\\\4t^2-2t-2t+1=0\\\\2t(2t-1)-1(2t-1)=0\\\\(2t-1)(2t-1)=0\\\\(2t-1)^2=0\iff2t-1=0\qquad\text{add 1 to both sides}\\\\2t=1\qquad\text{divide both sides by 2}\\\\t=(1)/(2)`

`\text{We're going back to substitution}\\\\t=(1)/(2)\Rightarrow2^x=(1)/(2)\qquad\text{use}\ a^(-1)=(1)/(a)\\\\2^x=2^(-1)\iff x=-1`