Problem 10.1. Consider the equations

$$
3^{2 x+3}-2^{x+2}=2^{x+5}-9^{x+1}
$$

and

$$
a .5^{2 x}+|a-1| 5^{x}=1
$$

where $a$ is a real number.

a) Solve the equation (1).

b) Find the values of $a$ such that the equations (1) and (2) are equivalent.

Kerope Chakarian
10.1. a) The equation (1) can be written as $9^{x}=2^{x}$, i.e. $\left(\frac{9}{2}\right)^{x}=1$ and $x=0$.

b) We plug the only solution $x=0$ of (1) in (2) and obtain $|a-1|=$ $1-a \Longleftrightarrow a \leq 1$. In this case 0 is a solution of (2) and we have to decide when (2) has no other solution(s).

Set $5^{x}=t, t>0$. Then we have to find all $a \leq 1$ such that the equation

$$
a t^{2}+(1-a) t-1=0
$$

has not positive roots different from 1 . For $a=0$ the only root of (3) is $t=1$ and the condition is satisfied. For $a \neq 0$ the roots of (3) are 1 and $-\frac{1}{a}$. Therefore the condition is satisfied if and only if $-\frac{1}{a}=1$ or $-\frac{1}{a}0$ and we conclude that $a \in[0,1] \cup\{-1\}$.