First slide

Theory of equations

Question

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

Moderate

A

$- 4, 1$
B

$4, - 1$
C

$2, \sqrt{3}$
D

$- 2, - \sqrt{3}$

Solution

One root is $2 + \sqrt{3},$ then second root must be $2 - \sqrt{3} .$
$\Rightarrow Sum of roots = - p = [2 + \sqrt{3} + 2 - \sqrt{3}] = 4 \Rightarrow p = - 4$
and product of roots $= q = (2 + \sqrt{3}) (2 - \sqrt{3}) = 4 - 3 = 1 .$
$\Rightarrow (p, q) = (- 4, 1)$

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