Question:
If $2+\sqrt{3}$ is root of the equation $x^{2}+p x+q=0$, than write the values of $p$ and $q$.
Solution:
Irrational roots always occur in conjugate pairs.
If $2+\sqrt{3}$ is a root and $2-\sqrt{3}$ is its conjugate root.
$\Rightarrow(2+\sqrt{3}+2-\sqrt{3})=-p$
$\Rightarrow 4=-9$
$\Rightarrow p=-4$
Also, $(2+\sqrt{3})(2-\sqrt{3})=q$
$\Rightarrow 4-3=q$
$\Rightarrow q=1$