Question

The values of ‘a’ for which $(a^2-1)x^2+2(a-1)x+2$ is positive for any $x$ are

• A. $a\ge 1$
• B. $a\le 1$
• C. $a$ < $-3$ or $a$ >$1$
• D. $a$ >$-3$

Answer 1

The correct option is C

$a$ < $-3$ or $a$ >$1$

We know that the expression $a{x}^2+bx+c>0$ for all $x$,
if $a>0$ and $b^2-4ac<0$

$\therefore (a^2–1)x^2+2(a–1)x+2$ is positive for all $x$ if

$a^2–1>0$ 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 $(a–1)(a+3)>0$

$\Rightarrow (a-1)(a+1)>0$ and
$a<–3$ or $a>1$

$\Rightarrow a<–3$ or $a>1$