First slide

Theory of equations

Question

The roots of the equation $a x^{2} + b x + c = 0, a \equiv 0, a, b, c \in R$ are non-real and $a + c < b,$ then

Moderate

A

$4 a + c = 2 b$
B

$4 a + c < 2 b$
C

$4 a + c > 2 b$
D

None of these

Solution

The given equation is $a x^{2} + b x + c = 0$
$a, b, c \in R$ , roots are non-real

$\Rightarrow$    $f (x) = a x^{2} + b x + c$ has same sign $\forall x \in R$

so $f (- 1) = a + c - b < 0$

$\Rightarrow$    $f (- 2) = 4 a - 2 b + c < 0$

$\Rightarrow$     $4 a + c < 2 b$

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