Question

Vă rog !!! Dau coroana !!!​

Answer 1

$\it Fie\ ecua\c{\it t}ia\ x^2-px+q=0, \ cu\ r\breve ad\breve acinile \ nenule\ x_1,\ x_2.\\ \\ Determina\c{\it t}i\ p,\ q\in\mathbb{R}\ astfel\ \hat\imath nc\hat at\ x_1,\ x_2,\ p,\ q\ s\breve a\ fie\ termeni\ consecuitivi\\ \\ ai\ unei\ progresii\ aritmetice\ .$

$\bf \underline{R}:$

$\it\ \div\ x_1,\ x_2,\ p,\ q\ \Rightarrow x_2=x_1+r,\ \ p=x_1+2r,\ \ q=x_1+3r\\ \\ x_1+x_2=p \Rightarrow x_1+x_1+r=x_1+2r \Rightarrow x_1=r \Rightarrow x_2=2r,\ p=3r,\ q=4r\\ \\ x_1\cdot x_2=q \Rightarrow r\cdot2r=4r \Rightarrow r=2\\ \\ Deci,\ x_1=2,\ \ x_2=2\cdot2=4,\ \ p=3\cdot2=6,\ \ w=4\cdot2=8$