2(square root 2)^2x+(square root 2)^x-1=0

Question

solve for x method?

Answer 1

2√2^2x+√2^x-1=0.

Let y=√2^x, then y²=√2^2x: 2y²+y-1=0=(2y-1)(y+1),

y=½ or -1, so √2^x=½=2^-1, 2^x/2=2^-1, so x/2=-1, x=-2. Note that √2^x cannot be negative.

CHECK: 2√2^2x=2×2^x=2^x+1=2^-1=½; √2^x=2^x/2=2^-1=½; ½+½-1=0✔️