Question

The equation ax^2+|2a-3|x-6 has no real roots if

Answer 1

Answer:

ax

2

+(a−2)x−2

The discriminant is (a−2)

2

−4(a)(−2)=a

2

−4a+4+8a=a

2

+4a+4=(a+2)

2

That's positive for all a, so there are always two roots. Now if a>0, then the quadratic is negative between those two real roots. Using the quadratic formula for roots :

2a

2−a−(a+2)

<×<

2a

2−a+a+2

−1<×<

a

2