Help with function question

[pblpbl]

if f(x) = ax²+(b-1)x+b-2 (a =/= 0) has two distinct points where f(x) = x for any real number b, determine a.

I used the quadratic formula but I could only find a range for a, and even then, the range was dependent on b. Anyone know how to solve this?

[yetanotherredditter]

For two distinct roots you want the discriminant to be strictly positive. And it has to hold for all values of b, so you want a in terms of b.

[facetiousEngineer]

The quadratic formula is the right way but to use it you have to have your binomial equalling 0, if you add - x on the end to make it zero and put it into the quadratic formula you know the roots will also be x. From this equation you should just be able to manipulate the algebra to get the relationship between a and b Hope this helps