1. **Stating the problem:** Solve the equation $$4^{x-3}\left(x-\frac{1}{2}\right) = 3^{x+\frac{1}{2}} - 2^{2x}$$ for $x$. 2. **Understanding the equation:** This is a transcendental equation involving exponential terms with different bases and polynomial factors. There is no straightforward algebraic method to isolate $x$. 3. **Approach:** We can try to simplify and analyze the equation or use numerical methods to approximate $x$. 4. **Rewrite the equation:** $$4^{x-3}\left(x-\frac{1}{2}\right) = 3^{x+\frac{1}{2}} - 2^{2x}$$ Recall that $4 = 2^2$, so: $$4^{x-3} = (2^2)^{x-3} = 2^{2(x-3)} = 2^{2x - 6}$$ Also, $2^{2x} = (2^2)^x = 4^x$. So the equation becomes: $$2^{2x - 6}\left(x - \frac{1}{2}\right) = 3^{x + \frac{1}{2}} - 4^x$$ 5. **Rewrite $3^{x + \frac{1}{2}}$ as:** $$3^{x + \frac{1}{2}} = 3^x \cdot 3^{\frac{1}{2}} = 3^x \sqrt{3}$$ 6. **Rewrite the equation:** $$2^{2x - 6}\left(x - \frac{1}{2}\right) = 3^x \sqrt{3} - 4^x$$ 7. **Numerical approximation:** Because of the complexity, we use numerical methods (like Newton-Raphson or graphing) to find $x$. 8. **Check for possible integer values:** Try $x=1$: Left side: $$2^{2(1) - 6} \left(1 - \frac{1}{2}\right) = 2^{-4} \cdot \frac{1}{2} = \frac{1}{16} \cdot \frac{1}{2} = \frac{1}{32} = 0.03125$$ Right side: $$3^1 \sqrt{3} - 4^1 = 3 \cdot 1.732 - 4 = 5.196 - 4 = 1.196$$ Not equal. Try $x=3$: Left side: $$2^{6 - 6} \left(3 - \frac{1}{2}\right) = 2^0 \cdot 2.5 = 1 \cdot 2.5 = 2.5$$ Right side: $$3^3 \sqrt{3} - 4^3 = 27 \cdot 1.732 - 64 = 46.764 - 64 = -17.236$$ Not equal. Try $x=2$: Left side: $$2^{4 - 6} \left(2 - \frac{1}{2}\right) = 2^{-2} \cdot 1.5 = \frac{1}{4} \cdot 1.5 = 0.375$$ Right side: $$3^2 \sqrt{3} - 4^2 = 9 \cdot 1.732 - 16 = 15.588 - 16 = -0.412$$ Closer but not equal. 9. **Conclusion:** The solution is approximately between $x=1$ and $x=2$. Using numerical methods (e.g., graphing or iterative approximation), the root can be found more precisely. **Final answer:** The exact solution cannot be expressed in elementary form; numerical approximation is required. Slug: "exponential equation" Subject: "algebra" Desmos: {"latex":"4^{x-3}(x-\frac{1}{2})=3^{x+\frac{1}{2}}-2^{2x}","features":{"intercepts":true,"extrema":true}} q_count:1