005 Sample Final A, Question 8

Question Solve the following equation,      $3^{2x}+3^{x}-2=0$

Foundations
1) What substitution can we make to simplify the problem?
Answer:
1) Substitute $y=3^{x}$ to change the original equation into $y^{2}+y-2=0$

Step 1:
Start by rewriting $3^{2x}=\left(3^{x}\right)^{2}$ and make the substitution $y=3^{x}$

Step 2:
After substitution we get $y^{2}+y-2=(y+2)(y-1)$

Step 3:
Now we have to find the zeros of $3^{x}+2=0$ and $3^{x}-1=0$. We do this by first isolating $3^{x}$ in both equations.
So $3^{x}=-2$ and $3^{x}=1$

Step 4:
We observe that $3^{x}=-2$ has no solutions. We can solve $3^{x}=1$ by taking $log_{3}$ of both sides.
This gives$\log _{3}\left(3^{x}\right)=x=\log _{3}(1)=0$

Final Answer:
x = 0