2. Find all solutions to the equation

$$
\sqrt[3]{x}+\sqrt[3]{2 x-3}=\sqrt[3]{12(x-1)}
$$
2. We will use the identity $(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b) . \quad 5$ points Cubing the given equation, we get:

$$
\begin{aligned}
x+(2 x-3)+3 \sqrt[3]{x} \cdot \sqrt[3]{2 x-3} \cdot(\sqrt[3]{x}+\sqrt[3]{2 x-3}) & =12(x-1) \\
3 \sqrt[3]{x} \cdot \sqrt[3]{2 x-3} \cdot \sqrt[3]{12(x-1)} & =9(x-1) \quad /: 3 /^{3} \\
x(2 x-3) \cdot 12 \cdot(x-1) & =27(x-1)^{3} \quad /: 3 \\
(x-1)\left[4 x(2 x-3)-9(x-1)^{2}\right] & =0 \\
(x-1)\left(-x^{2}+6 x-9\right) & =0 \\
(x-1)(x-3)^{2} & =0
\end{aligned}
$$

15 points

The solutions to the given equation are $x_{1}=1, x_{2,3}=3$.

5 points