Question:
The solution of the equation \( 4^{x} - 3 \cdot 2^{x+2} + 32 = 0 \) is:
The solution of the equation \( 4^{x} - 3 \cdot 2^{x+2} + 32 = 0 \) is:
Show Hint
Always rewrite terms like $2^{x+2}$ as $2^{2} \cdot 2^{x}$ to clearly see the quadratic structure.
Updated On: Apr 10, 2026
- $\{1, 2\}$
- $\{1, 3\}$
- $\{2, 3\}$
- $\{2, 4\}$
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The Correct Option is C
Solution and Explanation
Step 1: Concept
Convert the exponential equation into a quadratic equation by substitution.
Step 2: Analysis
Let $2^{x} = y$. Then $4^{x} = y^{2}$. The equation becomes: $y^{2} - 3(2^{2} \cdot y) + 32 = 0 \Rightarrow y^{2} - 12y + 32 = 0$. Factoring: $(y-8)(y-4) = 0$. So, $y = 8$ or $y = 4$.
Step 3: Conclusion
$2^{x} = 8 \Rightarrow x = 3$; $2^{x} = 4 \Rightarrow x = 2$. The solution set is $\{2, 3\}$.
Final Answer: (C)
Convert the exponential equation into a quadratic equation by substitution.
Step 2: Analysis
Let $2^{x} = y$. Then $4^{x} = y^{2}$. The equation becomes: $y^{2} - 3(2^{2} \cdot y) + 32 = 0 \Rightarrow y^{2} - 12y + 32 = 0$. Factoring: $(y-8)(y-4) = 0$. So, $y = 8$ or $y = 4$.
Step 3: Conclusion
$2^{x} = 8 \Rightarrow x = 3$; $2^{x} = 4 \Rightarrow x = 2$. The solution set is $\{2, 3\}$.
Final Answer: (C)
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