## Task B-2.1.

If the square and double the cube of a number are added, the result is three times the fourth power of the same number. Determine all real numbers for which this is true.
## Solution.

Let the required number be $x$. According to the conditions of the problem, we have $x^{2}+2 x^{3}=3 x^{4}$.

This equation is equivalent to $3 x^{4}-2 x^{3}-x^{2}=0$,

or $x^{2}\left(3 x^{2}-2 x-1\right)=0$.

1 point

It follows that $3 x^{2}-2 x-1=0$ or $x^{2}=0$. 1 point

Solving the quadratic equation yields two solutions $x_{1}=1, x_{2}=-\frac{1}{3}$. 2 points

and the third solution is $x_{3}=0$,

1 point

Note: If the student divides the equation by $x^{2}$ and thus loses the solution $x=0$, they lose 2 points, i.e., for two correct solutions with the process, they get 4 points.

If the student guesses one or two solutions without writing the equation, they get only 1 point. If they have the equation written and one or two guessed solutions, they get 3 points.