Answer to Question #236232 in Algebra for luka

Question

Use substitution y = 2^X to solve for X the equation 2^(2X +1)– 2^(X +1) + 1 = 2X

Answer 1

Let $y=2^x, y>0$

Substitute

$$2y^2-2y+1=y$$

$$2y^2-3y+1=0$$

$$y(2y-1)-(2y-1)=0$$

$$(2y-1)(y-1)=0$$

$$y_1=\dfrac{1}{2}, y_2=1$$

$$2^x=\dfrac{1}{2}$$

$$x=-1$$

$$2^x=1$$

$$x=0$$

$x_1=-1, x_2=0$

$\{-1, 0\}$