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

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Solution

Since, a root of the equation $x^{2} + p x + q = 0$ ...(i) If $2 + \sqrt{3}$ in one of the solution (root) of (i) in $2 - \sqrt{3}$ will be other roots Now, sum of roots = $\frac{- b}{a} \Rightarrow (2 + \sqrt{3}) + (2 - \sqrt{3}) = - p \Rightarrow 4 = - p a n d p r o d u c t o f r o o t = \frac{c}{a} \Rightarrow (2 + \sqrt{3}) (2 - \sqrt{3}) = 2 \Rightarrow 4 - 3 = q \Rightarrow q = 1 T h u s, p = - 4, q = 1$

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