First slide

Theory of equations

Question

$a x^{2} + b x + c = 0 (a \neq 0), a, b, c \in R,$ has no real root and $a + b + c < 0,$ then

Moderate

A

$c \leq 0$
B

$c > 0$
C

$c = 0$
D

$c < 0$

Solution

The given equation is $a x^{2} + b x + c = 0$
Given that $a x^{2} + b x + c = 0$ has no real root,

$f (x) = a x^{2} + b x + c$ has same sign for all real $x .$

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

$\Rightarrow$ $f (0) = c < 0$

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