Question

Se considera functia f:R-R,f(x)=2x^2-3x+5. Aratati ca exista un nr irational alfa pt care f(alfa) apartine la Q.

Answer 1

$f:\mathbb{R}\to\mathbb{R},\,\,\,\,f(x) = 2x^2-3x+5\\ \\ f(x) = (2x^2-3x-1)+6\\ \\ \text{Pentru }2x^2-3x-1:\\ \Delta = 3^2-4\cdot 2\cdot(-1) = 9+8 = 17\\ \Rightarrow x_{1,2} = \dfrac{3\pm \sqrt{17}}{4}\\ \\ \text{Fie }\alpha = \dfrac{3+\sqrt{17}}{4}\in \mathbb{I} \Rightarrow f(\alpha) = 0+6 \Rightarrow\\ \Rightarrow f\left(\dfrac{3+\sqrt{17}}{4}\right) = 6\in \mathbb{Q}$