4^x - 3^{x - frac{1}{2}} = 3^{x + frac{1}{2}} | vedclass

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(C) Given equation: $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$ Rearranging terms: $2^{2x} + 2^{2x - 1} = 3^{x + \frac{1}{2}} + 3^{

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$\frac{3}{2}$

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(C) Given equation: $4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1}$ Rearranging terms: $2^{2x} + 2^{2x - 1} = 3^{x + \frac{1}{2}} + 3^{x - \frac{1}{2}}$ Factor out common terms: $2^{2x}(1 + \frac{1}{2}) = 3^x(\sqrt{3} + \frac{1}{\sqrt{3}})$ Simplify: $2^{2x}(\frac{3}{2}) = 3^x(\frac{3 + 1}{\sqrt{3}}) = 3^x(\frac{4}{\sqrt{3}})$ Multiply both sides by $\sqrt{3}$: $2^{2x} \cdot 3 \cdot \sqrt{3} = 3^x \cdot 8$ Express in powers: $2^{2x} \cdot 3^{\frac{3}{2}} = 3^x \cdot 2^3$ Equating powers of $2$: $2x = 3 \Rightarrow x = \frac{3}{2}$ Equating powers of $3$: $x = \frac{3}{2}$ Thus,$x = \frac{3}{2}$.

$4^x - 3^{x - \frac{1}{2}} = 3^{x + \frac{1}{2}} - 2^{2x - 1} \Rightarrow x = $

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