If $(2 + i \sqrt{3})$ is a root of the equation $x^{2} + p x + q = 0$ where $p$ and $q$ are real, then find $(p, q)$ .

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Solution

As p and q are real and one root is $(2 + i \sqrt{3})$ , so the other root must be $(2 - i \sqrt{3})$ .

Then $p = - (s u m o f r o o t s) = - 4$

$q = p r o d u c t o f r o o t s = (2 + i \sqrt{3}) (2 - i \sqrt{3}) = 4 + 3 = 7$

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If $2 + i \sqrt{3}$ is a root of the equation $x^{2} + p x + q = 0$ , where p and q are real, then (p, q) =

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