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