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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3^{x}+2=0} and $3^{x}-1=0$. We do this by first isolating $3^{x}$ in both equations.
So Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3^{x}=-2} and $3^{x}=1$

Step 4:
We observe that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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