Quadratic EquationHard
Question
If the equation $ax^{2} + bx + c = 0$, where $a,b,c \in R$ and $a > 0$ has two real roots $\alpha$ and $\beta$ such that $\alpha < - 2$ and $\beta > 2$, then
Options
A.$c < 0$
B.$a - |b| + c < 0$
C.$4a + 2|\text{ }b| + c < 0$
D.$6a + |b| + c < 0$
Solution
Let $f(x) = ax^{2} + bx + c$
$$\begin{matrix} f( - 2) < 0 & \ \Rightarrow 4a - 2\text{ }b + c < 0 \\ & \ \Rightarrow \ 4a + c < 2\text{ }b \end{matrix}$$
and $f(2) < 0 \Rightarrow 4a + 2\text{ }b + c < 0$
$$\begin{matrix} & \Rightarrow & 4a + c < - 2b \\ \therefore & & 4a + c < - 2|b|. \end{matrix}$$
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