Question

The value 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. $a>-3$
• D. $a<-3$ or $a>1$

Answer 1

The correct option is D $a<-3$ or $a>1$

$y=a{x}^2+bx+c=\frac{(2ax+b{)}^2+4ac-b^2}{4a}$

Now it is clear, $y$ is positive when $4a$ is positive and $4ac-b^2$ is positive or $b^2-4ac<0$

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

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 $(a-1)(a+3)>0$

$\Rightarrow a^2>1$ and $a<-3$ or $a>1$

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