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

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Solution

roots of the given equation are $2 + i \sqrt{3}, 2 - i \sqrt{3}$ conjugate roots

$- p =$ sum of roots

$- p = (2 + i \sqrt{3}) + (2 - i \sqrt{3})$

$- p = 4$

$∴ p = - 4$

$q =$ product of roots

$q = (2 + i \sqrt{3}) \times (2 - i \sqrt{3})$

$q = 2^{2} - (i \sqrt{3})^{2}$

$q = 4 + 3 = 7$

$∴ q = 7$

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