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The quadratic equation $x^{2}+bx+c=0$ with real coefficients $b$ and $c$ , has a complex root $\sqrt 3 - i$ , then which of the following represents the value of $c-ib$ ?

Option 1)
$4-2\sqrt3 i$

Option 2)
$4+2\sqrt3 i$

Option 3)
$2\sqrt3 +4i$

Option 4)
$2\sqrt3 -4i$

Answers (1)

As learnt in

To form a Quadratic Equation given the roots -

$x^{2}-Sx+P= 0$

- wherein

S = Sum of roots

P = Product of roots

$x^{2}+bx+c=0$

$\Alpha$ $\alpha = \sqrt 3 -i$ ; $\beta = \sqrt 3 +i$

$-b=2 \sqrt 3\:\:\:\:\:\: \Rightarrow b=-2 \sqrt 3$

$c= \alpha \beta = (\sqrt3-i) (\sqrt3+i)=4$

$c-ib=4+2\sqrt 3 i$

Option 1)

$4-2\sqrt3 i$

This option is incorrect.

Option 2)

$4+2\sqrt3 i$

This option is correct.

Option 3)

$2\sqrt3 +4i$

This option is incorrect.

Option 4)

$2\sqrt3 -4i$

This option is incorrect.

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Vakul

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