Question

For $a\in Q-\{1,-2\}$, the roots of the equation $(a^2+a-2)x^2+(2a^2+a-3)x+a^2-1=0$ are

• A. irrational and equal
• B. irrational and distinct
• C. rational and equal
• D. rational and distinct

Answer 1

The correct option is D rational and distinct

$(a^2+a-2)x^2+(2a^2+a-3)x+a^2-1=0$
$\Delta =(2a^2+a-3{)}^2-4(a^2+a-2\left)\right(a^2-1)$
$=(2a+3{)}^2(a-1{)}^2-4(a-1)(a+2)(a^2-1)$
$=(a-1{)}^2[4a^2+9+12a-4(a^2+3a+2\left)\right]$
$=(a-1{)}^2>0\text{as}a\ne 1$

So, $\Delta$ is a perfect square and positive.
Hence, roots are rational and distinct.