I have this equation I have been working on and trying to decode it into the form of a quadratic equation of the form $$ax^2 + bx + c$$
The equation is following.
$$ 4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x-1}$$
And the value of x is $\frac{3}{2}$
I tried it like this after some time
$$ \Rightarrow 2^{2^x} + 2^{2x-1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}} $$
$$ \Rightarrow 2^{2x} + 2^{2x-1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}} $$
$$ \Rightarrow 2^{2x}( 1 +2^{-1}) = 3^x\left( 3^{\frac{1}{2}} + 3^{- \frac{1}{2}} \right)$$
$$ \Rightarrow 2^{2x}\left( 1 +\frac{1}{2}\right) = 3^x\left( \sqrt3 + \frac{1}{\sqrt3} \right)$$
$$ \Rightarrow 2^{2x} \left(\frac{3}{2}\right) = 3^x\left(\frac{4}{\sqrt3}\right)$$
$$ \Rightarrow \frac{2^{2x}}{3^x}=\frac{\left(\frac{4}{\sqrt3}\right)}{\left(\frac{3}{2}\right)}$$
$$ \Rightarrow \left(\frac{4}{3}\right)^x=\frac{\left(\frac{4}{\sqrt3}\right)}{\left(\frac{3}{2}\right)}$$
$$ \Rightarrow \left(\frac{4}{3}\right)^x=\left(\frac{4*2}{\sqrt3*3}\right)$$
$$ \Rightarrow \left(\frac{4}{3}\right)^x=\left(\frac{8}{3\sqrt3}\right)$$
$$ \Rightarrow 4^x=8$$ $$ \Rightarrow 2^{2x}=2^3$$ $$ \Rightarrow 2x = 3$$ $$ \Rightarrow x = \frac{3}{2}$$ $$ \Rightarrow 3^x=3\sqrt3$$ $$ \Rightarrow 3^x=3^1*3^\frac{1}{2}$$ $$ \Rightarrow 3^x=3^{1+\frac{1}{2}}$$ $$ \Rightarrow 3^x=3^{\frac{3}{2}}$$ $$ \Rightarrow x=\frac{3}{2}$$
And I succeeded
Thanks for this lol