Question
If $a ^2-3 a -1=0$ and $a \neq 0$, find $: a +\frac{1}{ a }$
✓
Answer
$a-\frac{1}{a}=3$
Squaring both sides, we get
$\left(a-\frac{1}{a}\right)^2 $
$=a^2+\frac{1}{a^2}-2 $
$=9 $
$=a^2+\frac{1}{a^2} $
$=11 .$
Now,
$\left(a+\frac{1}{a}\right)^2 $
$=a^2+\frac{1}{a^2} $
$=11+2 $
$=13 $
$\Rightarrow a+\frac{1}{a^2} $
$= \pm \sqrt{13} .$
Squaring both sides, we get
$\left(a-\frac{1}{a}\right)^2 $
$=a^2+\frac{1}{a^2}-2 $
$=9 $
$=a^2+\frac{1}{a^2} $
$=11 .$
Now,
$\left(a+\frac{1}{a}\right)^2 $
$=a^2+\frac{1}{a^2} $
$=11+2 $
$=13 $
$\Rightarrow a+\frac{1}{a^2} $
$= \pm \sqrt{13} .$
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