# 005 Sample Final A, Question 8

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

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

Step 1:
Start by rewriting ${\displaystyle 3^{2x}=\left(3^{x}\right)^{2}}$ and make the substitution ${\displaystyle y=3^{x}}$
Step 2:
After substitution we get ${\displaystyle y^{2}+y-2=(y+2)(y-1)}$
Step 3:
Now we have to find the zeros of ${\displaystyle 3^{x}+2=0}$ and ${\displaystyle 3^{x}-1=0}$. We do this by first isolating ${\displaystyle 3^{x}}$ in both equations.
So ${\displaystyle 3^{x}=-2}$ and ${\displaystyle 3^{x}=1}$
Step 4:
We observe that ${\displaystyle 3^{x}=-2}$ has no solutions. We can solve ${\displaystyle 3^{x}=1}$ by taking ${\displaystyle log_{3}}$ of both sides.
This gives ${\displaystyle \log _{3}\left(3^{x}\right)=x=\log _{3}(1)=0}$