First slide

Theory of expressions

Question

The values of 'a' for which $(a^{2} - 1) x^{2} + 2 (a - 1) x + 2$ is positive for any x, are

Moderate

A

$a \geq 1$
B

$a \leq 1$
C

$a > - 3$
D

$a < - 3 or a > 1$

Solution

We know that, the expression ${ax}^{2} + bx + c > 0$ for all x, if a >0 and b² < 4ac
$∴ (a^{2} - 1) x^{2} + 2 (a - 1) x + 2$ is positive for all x, if
$\begin{array}{l} a^{2} - 1 > 0 and 4 (a - 1)^{2} - 8 (a^{2} - 1) < 0 \\ \Rightarrow a^{2} - 1 > 0 and - 4 (a - 1) (a + 3) < 0 \\ \Rightarrow a^{2} - 1 > 0 and (a - 1) (a + 3) > 0 \\ \Rightarrow a^{2} > 1 and a < - 3 or a > 1 \\ \Rightarrow a < - 3 or a > 1 \end{array}$

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