SUBIECTUL 2

a) If $x$ is a real number and $x(x+1)=12$, calculate the value of the expression $E(x)=x^{4}+2 x^{3}+2 x^{2}+x$.

b) Show that $A=\sqrt{a^{4}+2 a^{3}+2 a^{2}+a}$ is irrational for any non-zero natural number $a$.

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SUBIECTUL 2

a) If $x$ is a real number and $x(x+1)=12$, calculate the value of the expression $E(x)=x^{4}+2 x^{3}+2 x^{2}+x$.

b) Show that $A=\sqrt{a^{4}+2 a^{3}+2 a^{2}+a}$ is irrational for any non-zero natural number $a$.

| a) $E(x)=x^{4}+x^{3}+x^{3}+x^{2}+x^{2}+x=x^{3}(x+1)+x^{2}(x+1)+x(x+1)$ (or another grouping) |  |
| :---: | :---: |
| $=x(x+1)\left(x^{2}+x+1\right)=12(12+1)=156$ | ![](https://cdn.mathpix.com/cropped/2024_06_07_2543a31e59990b7d1ba3g-2.jpg?height=55&width=92&top_left_y=799&top_left_x=1881) |
| that solves the equation $x^{2}+x-12=0 (1 p)$ and 1 p for each substitution. |  |
| b) According to the factorization from part a) we have $\mathrm{a}^{4}+2 a^{3}+2 a^{2}+a=\left(a^{2}+a\right)\left(a^{2}+a+1\right)$. | 2p |
| If $a^{2}+a=n$, then $n^{2}$ is a perfect square since it lies between two consecutive perfect squares. |  |