Question

If the expression $(a-2)x^2+2(2a-3)x+(5a-6)$ is positive for all real values of $x$, then

• A. $a$ can be any real number
• B. $a>1$
• C. $a>3$
• D. $a=3$

Answer 1

The correct option is C $a>3$

The expression is always positive that means the discriminant of the equation is always less than zero .
So, $4(2a-3{)}^2-4(a-2\left)\right(5a-6)<0$
$(16a^2+36-48a)-(20a^2-64a+48)<0$
$(-4a^2+16a-12)<0$
$(-4)(a^2-4a+3)<0$
$(a-3)(a-1)>0$
So, the expression will be positive for $a>3$ and $a<1$
Option C is correct .