Answer:
x = -1
Step-by-step explanation:
[tex]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=\dfrac{1}{2}[/tex]
[tex]\text{We're going back to substitution}\\\\t=\dfrac{1}{2}\Rightarrow2^x=\dfrac{1}{2}\qquad\text{use}\ a^{-1}=\dfrac{1}{a}\\\\2^x=2^{-1}\iff x=-1[/tex]
