If one root of the equation $a x^{2} + b x + c = 0$ is $3 - 4 i$ , then $a + b + c$ is equal to

40

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36

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- 20

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20

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Solution

The correct option is C $20$
Since $a, b, c$ are all real, hence the roots occur is conjugate pair.
Hence if one root is $3 - 4 i$ then the other root is $3 + 4 i$ .
Thus
$\frac{- b}{a} = (3 + 4 i) + (3 - 4 i)$
$= 6$ ...(i)
And
$\frac{c}{a} = (3 - 4 i) (3 + 4 i) = 25$ .
Hence
$a x^{2} + b x + c = 0$
$x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$
$x^{2} - 6 x + 25 = 0$
Therefore
$c = 25,$ $b = - 6$ and $a = 1$ .
Hence
$a + b + c$
$= 25 - 6 + 1$
$= 25 - 5$
$= 20$ .

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