First slide

Theory of expressions

Question

If $a, b, c \in R$ and the quadratic equation $x^{2} + (a + b) x + c = 0$ has no real roots, then

Moderate

A

$c (a + b + c) > 0$
B

$c + (a + b + c) c > 0$
C

$c - c (a + b + c) > 0$
D

$c (a + b - c) > 0$

Solution

We have,
$x^{2} + (a + b) x + c = 0$ …(i)
Let $f (x) = x^{2} + (a + b) x + c$
Since coefficient of $x^{2}$ is positive. So, the equation (i) will not have real roots, if
$f (x) > 0$ for all $x \in R .$
$\Rightarrow f (0) f (1) > 0$ and $f (0) f (- 1) > 0$
$\Rightarrow c (1 + a + b + c) > 0$ and $c (1 - a - b + c) > 0$
$\Rightarrow c + c (a + b + c) > 0$ and $c - c (a + b - c) > 0$
Hence, options $(b)$ is are true.

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