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Q.
The value of $'a'$ for which $(a^2 - 1) x + 2 (a - 1) x + 2$ is positive for any $x$, are
Complex Numbers and Quadratic Equations
Solution:
We have that $ax^2 + bx + c > 0$ for all $x$ if $a$ > 0 and $b^2 < 4\, ac$
$ \therefore \, (a^2 - 1)x^2 + 2 (a - 1) x + 2$ is +ve for all $x$ if
$a^2-1>0$ and $4(a-1)^2 - 8 (a^2-1)\leq 0$
$\Rightarrow \, a^2 - 1 \geq \, 0$ and $- 4 (a - 1) (a + 3) \leq \, 0 $
$\Rightarrow \, a^2 - 1 \geq $ and $(a - 1) (a + 3) \geq 0$
$\Rightarrow \, a \leq - 1$ or $a \geq $ and $a \leq - 3 $ or $a \geq 1$
$\Rightarrow \, a \leq - 3 $ or $a \geq 1$