Question

If $2+\sqrt{3}$ is root of the equation $x^2+px+q=0$, than write the values of p and q.

Answer 1

Irrational roots always occur in conjugate pairs.
If $$\begin{gathered} 2+\sqrt{3}\mathrm{is}\mathrm{a}\mathrm{root}\mathrm{and}2-\sqrt{3}\mathrm{is}\mathrm{its}\mathrm{conjugate}\mathrm{root}. \\ \Rightarrow \left(2+\sqrt{3}+2-\sqrt{3}\right)=-p \\ \Rightarrow 4=-9 \\ \Rightarrow p=-4 \\ \mathrm{Also},\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)=q \\ \Rightarrow 4-3=q \\ \Rightarrow q=1 \end{gathered}$$