If $c^{2} = 4 d$ and the two equations $x^{2} - a x + b = 0$ and $x^{2} - c x + d = 0$ have one common root, then the value of $2 (b + d)$ is equal to

\frac{a}{c}

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a c

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2 a c

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a + c

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Solution

The correct option is B $a c$
We can observe that the discriminant of the second equation is 0.
Hence, the root is $\frac{c}{2}$ which is also the common root.
Substituting the value of the common root in the first equation we get,
$b = \frac{a c}{2} - \frac{c^{2}}{4}$
Substituting the value of $d$ , we get,
$b + d = \frac{a c}{2}$
Hence, B is the correct option.

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