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If 2+i and ...

Question

If (2+i) and $(\sqrt{5} - 2 i)$ are the roots of the. equation $(x^{2} + a x - + b) (x^{2} + c x + d) = 0$ , where $a, b, c$ and d are real constants, then product of all the roots of the equation is

40

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9 \sqrt{5}

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45

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35

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Solution

The correct option is B $45$
If one root of a quadratic equation is of the form $a + i b$ , then other root will be $a - i b$ .
So, all the roots are $2 \pm i$ , $\sqrt{5} \pm 2 i$ .
$∴$ Product of all the roots
$= (2 + i) (2 - i) (\sqrt{5} + 2 i) (\sqrt{5} - 2 i)$
$= (4 + 1) (5 + 4) = 5 \times 9 = 45$

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