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 ≥ 1
• B. a ≤ 1
• C. a > -3
• D. a < -3 or a > 1

Answer 1

The correct option is D

a < -3 or a > 1

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

∴ $(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

⇒ $a^2-1$ > 0 and -4(a - 1)(a + 3) < 0

⇒ $a^2-1$ > 0 and (a - 1)(a + 3) > 0

⇒ $a^2$ > 1 and a < -3 or a > 1

⇒ a < -3 or a > 1