Question

Solve the equation $\sqrt{a(2^x-2)+1}=1-2^x$ for every value of the parameter $a$.

Answer 1

$\sqrt{a(2^x-2)+1}=1-2^x$

Squaring both sides, we have

${\left(\sqrt{a(2^x-2)+1}\right)}^2={(1-2^x)}^2$

$2^{x}a-2a+1=1+2^{2x}-2.2^x$

$a(2^x-2)=2^x(2^x-2)$

$\Rightarrow a=2^x$