Show $2^x-2^{-x}=1$ can be written as a quadratic equation

Question

I tried to solve this but i can't find the answer to know if i'm right or wrong

i) Show that if $y=2^x$ then the equation $2^x-2^{-x}= 1$ can be written as a quadratic equation in $y$.

ii) Hence solve the equation $2^x-2^{-x}=1$.

Answer 1

$2^x-2^{-x}=2^x-(2^x)^{-1}=y-y^{-1}=1$ Multiply both sides by $y$ you get: $y^2-1=y$.

Answer 2

Hint:

If $y=2^x$, then $;2^{-x}=\dfrac 1y$.

Answer 3

Put $$2^x = y$$

Then $$2^{-x} = \frac 1{2^x}$$

$$= \frac 1y$$

Then equation,

$$2^x - 2^{-x} = 1$$

$$y - \frac 1y = 1$$

$$\frac {y^2 - 1}{y} = 1$$

$$y^2 - 1 = y$$

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

Hope you can solve this quadratic equation further.