Which of the following is always true

Name: Interaction with AIR1 JEE Advanced 2023 | Chidvilasreddy JEE Topper
Uploaded: 2023-05-17T14:00:00.000Z
Duration: 20 min 32 s
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A
One root of the equation $a x^{2} + b x + c = 0, (a, b, c \in R & a \neq 0)$ in the form of $p + \sqrt{q}$ then other root is $p - \sqrt{q}$
B
Exactly one of the root of the equation $a x^{2} + b x + c = 0, (a \neq 0)$ lies in the given interval $(k_{1}, k_{2})$ if $f (k_{1}) . f (k_{2}) > 0$
C
If $a, b, c, d, e, f$ are positive real numbers and $d f - e^{2} > 0$ .
The equations $a x^{2} + 2 b x + c = 0 (a \neq 0)$ and
$d x^{2} + 2 e x + f = 0 (d \neq 0)$ have a common root,
then $\frac{d}{a} = \frac{e}{b} = \frac{f}{c}$
D
Equation $(a x^{2} + b x + c) (a x^{2} - d x - c) = 0, (a c \neq 0)$ has no real roots,

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