Question
Form the quadratic equation from the roots given below : $2-\sqrt{5}, 2+\sqrt{5}$
✓
Answer
Let $\alpha=2-\sqrt{5}$ and $\beta=2+\sqrt{5}$
$\therefore \alpha+\beta=2-\sqrt{5}+2+\sqrt{5}=4 \text { and } \alpha \beta=(2-\sqrt{5})(2+\sqrt{5})=4-5=1$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\begin{array}{l}
\therefore x ^2-(4) x +(1)=0 \\
\therefore x ^2-4 x +1=0
\end{array}$
$\therefore \alpha+\beta=2-\sqrt{5}+2+\sqrt{5}=4 \text { and } \alpha \beta=(2-\sqrt{5})(2+\sqrt{5})=4-5=1$
$\therefore$ and quadratic equation is, $x ^2-(\alpha+\beta) x +\alpha \beta=0$
$\begin{array}{l}
\therefore x ^2-(4) x +(1)=0 \\
\therefore x ^2-4 x +1=0
\end{array}$
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