The values of a for which (a^2 - 1)x^2 + 2(a | vedclass

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(D) For a quadratic expression $Ax^2 + Bx + C$ to be positive for all $x \in \mathbb{R}$,two conditions must be satisfied: $1$. The coefficient of $x^

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$a  1$

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(D) For a quadratic expression $Ax^2 + Bx + C$ to be positive for all $x \in \mathbb{R}$,two conditions must be satisfied: $1$. The coefficient of $x^2$ must be positive,i.e.,$A > 0$. $2$. The discriminant $D = B^2 - 4AC$ must be negative,i.e.,$D Given the expression $(a^2 - 1)x^2 + 2(a - 1)x + 2 > 0$: Condition $1$: $a^2 - 1 > 0 \implies a^2 > 1 \implies a > 1$ or $a Condition $2$: $D = [2(a - 1)]^2 - 4(a^2 - 1)(2) $4(a - 1)^2 - 8(a^2 - 1) Dividing by $4$: $(a - 1)^2 - 2(a^2 - 1) $(a^2 - 2a + 1) - 2a^2 + 2 $-a^2 - 2a + 3 Multiplying by $-1$ (reversing the inequality): $a^2 + 2a - 3 > 0$. $(a + 3)(a - 1) > 0$. This inequality holds when $a > 1$ or $a Taking the intersection of Condition $1$ ($a > 1$ or $a  1$ or $a The common values are $a > 1$ or $a < -3$.

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

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