First slide

Theory of expressions

Question

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ such that $α < - 1$ and $β > 1$ , then $1 + \frac{c}{a} + |\frac{b}{a}|, i s$

Moderate

A

$< 0$
B

$> 0$
C

$\leq 0$
D

none of these

Solution

Let $f (x) = a x^{2} + b x + c$ . Clearly, $- 1$ and $1$ lie between the roots of $f (x) = 0 .$
$∴ a f (1) < 0$ and $a f (- 1) < 0$
$\Rightarrow a (a + b + c) < 0$ and $a (a - b + c) < 0$
$\Rightarrow a^{2} (1 + \frac{b}{a} + \frac{c}{a}) < 0$ and $a^{2} (1 - \frac{b}{a} + \frac{c}{a}) < 0$
$\Rightarrow 1 + \frac{b}{a} + \frac{c}{a} < 0$ and $1 - \frac{b}{a} + \frac{c}{a} < 0 \Rightarrow 1 + |\frac{b}{a}| + \frac{c}{a} < 0$

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