005 Sample Final A, Question 8

Question Solve the following equation,      Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3^{2x}+3^{x}-2=0}

Foundations
1) What substitution can we make to simplify the problem?
Answer:
1) Substitute y = Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3^{x}} in both equations.
So $3^{x}=-2$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = 1}

Step 4:
We observe that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = -2} has no solutions. We can solve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = 1} by taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_3} of both sides.
This givesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_3\left(3^x\right) = x = \log_3(1) = 0}

Final Answer:
x = 0