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Numere rationale si irationale 1
Posted: Fri Dec 19, 2008 4:23 pm
by Marius Mainea
a) Fie x un numar real astfel incat \( x^2+x \) si \( x^3+2x \) sa fie rationale. Aratati ca x este numar rational.
b) Aratati ca exista numere irationale x astfel incat \( x^2+x \) si \( x^3-2x \) sa fie irationale.
Posted: Mon Dec 29, 2008 7:49 pm
by Claudiu Mindrila
Solutie.
\( a) \) \( x^{2}+x\in\mathbb{Q},x^{3}+2x\in\mathbb{Q}\Rightarrow x^{2}\left(x+1\right)+3x\in\mathbb{Q}\Rightarrow x\left[x\left(x+1\right)+3\right]\in\mathbb{Q}\Rightarrow x\left(a+3\right)\in\mathbb{Q},a+3\in\mathbb{Q}\Rightarrow x\in\mathbb{Q}. \)
\( b) \) Cu notatiile \( a=x^{2}+x\in\mathbb{Q},c=x^{3}-2x\in\mathbb{Q} \) avem \( ax-a+3x=c\Leftrightarrow x\left(a-3\right)=a+c\in\mathbb{Q},x\in\mathbb{R}-\mathbb{Q}\Leftrightarrow a-3=0\Leftrightarrow a=3. \).
Acum \( x^{2}+x=3\Leftrightarrow x^{2}+x-3=0. \). Cum \( \Delta=1-4\cdot\left(-3\right)=1+12=13>0 \), ecuatia are doua radacini distincte, anume \( x_{1,2}=\frac{-1\pm \sqrt{13}}{2} \) cu \( x_1,x_2\in \mathbb{R} -\mathbb{Q} \).