quadratic equation problem.

Question

I have been fumbling for months now to acquire the solution. My previous question was unclear, sorry for that.

$$\large 2^{x/2}+3^{x/2}=13^{x/4}$$

Please try to solve it.

Thanks in advance!!!

Answer 1

Our question:

To find $t$, such that $4^t+9^t=13^t$

$(\frac{4}{13})^t+(\frac{9}{13})^t=1$. Note that $t=1$ is trivial solution.

$\frac{4}{13},\frac{9}{13}<1$. Then, if $t>1$, $(\frac{4}{13})^t<\frac{4}{13}$. Similarly $(\frac{9}{13})^t<\frac{9}{13}$. So $(\frac{4}{13})^t+(\frac{9}{13})^t<(\frac{4}{13})+(\frac{9}{13})=1$.(Why?)

if $0<t<1$, $(\frac{4}{13})^t>\frac{4}{13}$. Similarly $(\frac{9}{13})^t>\frac{9}{13}$. So $(\frac{4}{13})^t+(\frac{9}{13})^t>(\frac{4}{13})+(\frac{9}{13})=1$.(Why?)

Hence $t=1$ is only solution.

Note: If you do not understand "Why" part, comment please. I will edit the post.

Answer 2

HINT: write your equation in the form $$2^{x/2}+3^{x/2}=13^{x/4}$$ and $$4^t+9^t=13^t$$ from the hint above!

Answer 3

Hint:  by inspection $x=4$ is a solution, then show that $\displaystyle\left(\frac{2}{\sqrt{13}}\right)^{x/2}+\left(\frac{3}{\sqrt{13}}\right)^{x/2}$ is strictly decreasing, so $x=4$ is the only solution.