Question

x+1/x=6
determinati x-1/x=?

Answer 1

Ridicam la patrat

x^2 + (1/x)^2 +2 = 36 ⇒ x^2 + (1/x)^2 = 34

(x- 1/x)^2 = x^2 + (1/x)^2 - 2 = 34-2= 32 = 16*2 si extragand radicalul ⇒

x- 1/x = +- 4rad2

Answer 2

$\it Not\breve{a}m\ x=a,\ \ \dfrac{1}{x}=b,\ cu\ condi\c{\it t}ia\ evident\breve{a}\ ab=1\ \ \ \ (*).\\ \\ \\ Rela\c{\it t}ia\ dat\breve{a}\ devine:\\ \\ \\ a+b=6 \Rightarrow (a+b)^2=6^2\Rightarrow a^2+2ab+b^2=36 \stackrel{(*)}{\Longrightarrow}a^2+2+b^2=36|_{-2} \Rightarrow\\ \\ \\ \Rightarrow a^2+b^2=34|_{-2ab} \Rightarrow a^2+b^2-2ab=34-2ab \stackrel{(*)}{\Longrightarrow} (a-b)^2 =32 \Rightarrow$

$\it \Rightarrow \sqrt{(a-b)^2}=\sqrt{32} \Rightarrow \sqrt{(a-b)^2}=\sqrt{16\cdot2} \Rightarrow \sqrt{(a-b)^2}=4\sqrt{2} \Rightarrow \\ \\ \\ \Rightarrow |a-b| = 4\sqrt2 \Rightarrow a-b=\pm4\sqrt2.$

Revenind asupra notației, rezultă:

$\it x-\dfrac{1}{x}=\pm4\sqrt2$