Solving $4\cdot 25^x-25\cdot 4^{x+1}=9\cdot 10^x$

Question

Solve the following equation:$$4\cdot 25^x-25\cdot 4^{x+1}=9\cdot 10^x.$$

We can decompose it to

$$2^2\cdot 5^{2x}-5^2\cdot 2^{2(x+1)}=3^2\cdot 10^x$$

$$2^2\cdot 5^{2x}-5^2\cdot 2^{2}\cdot 2^{2x}=3^2\cdot 10^x$$

How can we continue from here?

Answer 1

Note that $10^x = 2^x\cdot 5^x$. Divide by $2^{2x}$ on both sides, and you now have a quadratic equation in the unknown $(5/2)^x$.

Answer 2

Let $\left(\frac{5}{2}\right)^x=t$.

Hence, $$4t^2-100=9t$$ or $$\left(2t-\frac{9}{4}\right)^2=100+\frac{81}{16}$$ or $$\left(2t-\frac{9}{4}\right)^2=\left(\frac{41}{4}\right)^2,$$ which gives $t=\frac{25}{4}$ and $x=2$.