Question

a + 1/a = $\sqrt[]{13}$ sa se arate ca a - 1/a = 3

Answer 1

Explicație pas cu pas:

a+1/a=V13 /^2=> a²+2*a*1/a+1/a²=13=>a²+2+1/a²=13=>

a²+1/a²=11

a²+1/a²=11=>a²-2+1/a²=9=>(a-1/a)²=9 => a-1/a=3 sau

a-1/a=-3

Answer 2

$a+\dfrac{1}{a} = \sqrt{13} \Big|^2\\ \\\Leftrightarrow\, a^2+2\cdot a\cdot \dfrac{1}{a}+\dfrac{1}{a^2}=13\\ \\\Leftrightarrow\,a^2+2+\dfrac{1}{a^2}=13\\ \\ \Leftrightarrow\,a^2+\dfrac{1}{a^2} = 11\\ \\ \Leftrightarrow\,a^2-2+\dfrac{1}{a^2} = 11-2\\ \\ \Leftrightarrow\,a^2-2\cdot a\cdot \dfrac{1}{a}+\dfrac{1}{a^2} = 9\\ \\ \Leftrightarrow\,\left(a-\dfrac{1}{a}\right)^2 = 9\\ \\ \Leftrightarrow\, a-\dfrac{1}{a} = \pm \sqrt{9}\\ \\ \Leftrightarrow\,a-\dfrac{1}{a} = \pm 3$

$\Rightarrow \,\boxed{a-\dfrac{1}{a} = 3}$