First slide

Theory of expressions

Question

Given that $a x^{2} + b x + c = 0$ has no real roots and $a + b + c < 0$ then

Moderate

A

$c = 0$
B

$c > 0$
C

$c < 0$
D

$c = 0$

Solution

Since the equation $a x^{2} + b x + c = 0$ has no real roots. Therefore, the curve $y = a x^{2} + b x + c$ does not intersect with x-axis. Consequently $ϕ (x) = a x^{2} + b x + c$ c has same sign for all values of $x .$ It is given that
$\begin{array}{l} a + b + c < 0 \\ \Rightarrow ϕ (1) = a + b + c < 0 \end{array}$
$\Rightarrow ϕ (x) < 0$ for all $x \Rightarrow ϕ (0) < 0 \Rightarrow c < 0$

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