Assertion (A): If $2 x^{2} + 3 x + 4 = 0$ and $a x^{2} + b x + c = 0$ have a common root, then $a : b : c = 2 : 3 : 4$ ( $a, b$ and $c$ are real numbers) .
Reason (R): For a quadratic equation in $x$ with real coefficients, complex roots occur in conjugate pairs.

Both A and R are true and R is a correct explanation of A.

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Both A and R are true but R is not a correct explanation of A.

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A is true but R is false.

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Both A and R are false.

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Solution

The correct option is B Both A and R are true and R is a correct explanation of A.
$2 x^{2} + 3 x + 4 = 0$
$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 + 2}$
$∴$ Roots of the quadratic are complex.
$∴$ Both root are common in the equations $2 x^{2} + 3 x + 4 = 0$ and $a x^{2} + b x + c = 0$ .
Because complex roots occur in pairs.
$∴ \frac{a}{2} = \frac{b}{3} = \frac{c}{4}$
$∴ a : b : c = 2 : 3 : 4$

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