Solving for $x$ in: $4 \cdot 3^x - 9 \cdot 2^x = 5 \cdot 3^\frac x2 \cdot 2^ \frac x2$

Question

$$4 \cdot 3^x - 9 \cdot 2^x = 5 \cdot 3^\frac x2 \cdot 2^ \frac x2$$

How to solve this equality for x?

Answer 1

The hint.

Use the following substitution.

$$\left(\frac{3}{2}\right)^{\frac{x}{2}}=t$$

Answer 2

dividing by $3^{x/2}$ we get $$4\cdot 3^{x/2}-9\frac{2^x}{3^{x/2}}=5\cdot 2^{x/2}$$ dividing by $2^{x/2}$ we get $$4\cdot \left(\frac{3}{2}\right)^{x/2}-9\cdot \left(\frac{2}{3}\right)^{x/2}=5$$ Setting $$u=\left(\frac{3}{2}\right)^{x/2}$$ then we get $$4u-\frac{9}{u}=5$$ can you finish?