Question
Form the quadratic equation whose root are:
$2+\sqrt{5}$ and $2-\sqrt{5}$
$2+\sqrt{5}$ and $2-\sqrt{5}$
✓
Answer
Let $a, \beta$ be the given roots.
$\text { Then } a=2+\sqrt{5} \text { and } \beta=2-\sqrt{5} $
$a+\beta=2+\sqrt{5}+2-\sqrt{5}=4$
$ \text { and } a \beta=(2+\sqrt{5})(2-\sqrt{5})$
$\Rightarrow a+\beta=4 \text { and } a \beta=(2)^2-(\sqrt{5})^2$
$ \Rightarrow a+\beta=4 \text { and } a \beta=4-5$
$ \Rightarrow a+\beta=4 \text { and } a \beta=-1$
Required quadratic equation
$x^2-(a+\beta) x+a \beta=0$
$ \Rightarrow x^2-4 x-1=0$
$\text { Then } a=2+\sqrt{5} \text { and } \beta=2-\sqrt{5} $
$a+\beta=2+\sqrt{5}+2-\sqrt{5}=4$
$ \text { and } a \beta=(2+\sqrt{5})(2-\sqrt{5})$
$\Rightarrow a+\beta=4 \text { and } a \beta=(2)^2-(\sqrt{5})^2$
$ \Rightarrow a+\beta=4 \text { and } a \beta=4-5$
$ \Rightarrow a+\beta=4 \text { and } a \beta=-1$
Required quadratic equation
$x^2-(a+\beta) x+a \beta=0$
$ \Rightarrow x^2-4 x-1=0$
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