Solving this equation

Question

Question:

Solve: $$3^{2x^2}-2\cdot3^{x^2+x+6}+3^{2(x+6)}=0$$

I thought that we can take $a=3^{x^2}$ and $b = 3^{x+6}$. Then equation becomes $a^2-2ab+b^2=0$, which obviously means $a-b=0$. Substituing, I got:

$$3^{x^2}-3^{x+6}=0$$

But, I can't proceed further. I have an intuition that this is quadratic equation, but can't figure out how to. Please help me on this.

Thanks.

Answer 1

If $3^{x} = 3^y$, then $x=y$. In your case, you have $3^{x^2} = 3^{x+6}$.