Solve $a^{2x}+ a^4 = a^{x+1}+ a^{x+3}$

Question

Solve for the values of $x:$

$$a^{2x}+ a^4 = a^{x+1}+ a^{x+3}.$$

My attempt has been to make the base the same so I can cancel and add exponents:

$a^x.a^x+ a^4 = a^x.a^1 + a^x.a^3$

$a^x.a^x+ a^4 = a^x + a^x + a^4$

$a^x.a^x= a^x + a^x$

$a^x.a^x= 2a^x$

Now I'm stuck and need guidance on the next steps.

Answer 1

Assume $a^x$ as $p$ and you will arrive at a quadratic. $p^2-p(a+a^3)+a^4=0$. The values of $p$ are $a^3$ and $a$. Substituting $p$ as $a^x$, we get $x=\{1, 3\}$.

Answer 2

Hint: your initial equation is equivalent to $$(a^x-a^3)(a^x-a)=0.$$