TITLE: Solve equation $1-3^{x-1}+2^{\frac{x}{2}}-2^{\frac{x}{2}}3^{\frac{x-1}{2}}=0$
QUESTION [0 upvotes]: For $x \in R$, solve $$1-3^{x-1}+2^{\frac{x}{2}}-2^{\frac{x}{2}}3^{\frac{x-1}{2}}=0$$.
I've tried to notate $a=2^{\frac{x}{2}}, b=3^{\frac{x}{2}}$, but gotten nowhere. I've also tried to divide by $2^{\frac{x}{2}}\cdot3^{\frac{x}{2}}$, but the result is more or less the säme as the initial equation.

REPLY [4 votes]: Set $2^{x/2}=b, 3^{(x-1)/2}=a$
$$0=1-a^2+b-ab=(1-a)(1+a)+b(1-a)=(1-a)(1+a+b)$$
For real $x, a,b>0\implies a+b+1>1\ne0$
Now $3^{(x-1)/2}=a=1\implies(x-1)/2=0$
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