First slide

Theory of expressions

Question

If the equation ax² + bx + c = 0 (a > 0) has two real roots $α and β$ such that $α < - 2 and β > 2$ , then which of the following statements is/are true?

Moderate

A

$a - | b | + c < 0$
B

$c < 0, b^{2} - 4 ac > 0$
C

$4 a - 2 | b | + c < 0$
D

$9 a - 3 | b | + c < 0$

Solution

$\begin{array}{l} f (x) = {ax}^{2} + bx + c \\ f (0) = c < 0, D > 0 \\ \Rightarrow b^{2} - 4 ac > 0 \\ f (1) < 0 and f (- 1) < 0 \\ \Rightarrow a - | b | + c < 0 \\ f (2) < 0 and f (- 2) < 0 \\ \Rightarrow 4 a - 2 | b | + c < 0 \end{array}$
Nothing can be said about f(3) or f(-3), whether it is positive or negative.

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