The question below will involve spoilers.

I would really appreciate it if someone could help me understand the solution for the second part of the STEP question. I completed these assignments about a year ago but thought that I should revisit this as revision before starting university and now that I have read more about necessary and sufficient conditions I think I may be overthinking it.

Would this be an appopriate method?

• Calculate an expression for the discriminant of the quadratic.
• Substitute the given c^2 into the discriminant.
• Show that the discriminant evaluates to 0.
• Show that the solution is not equal to a or b (though I failed to do this).

I suggest the above method as from what I understand the question is to show that c^2 =... is a sufficient condition for the equation in question to have exactly one solution.

The worked solution starts with the fact that the discriminant equals 0 and shows that c^2 must be what is given in the question. I am confused as to whether this answers the question - the problem is probably not the answer (I am pretty sure I am wrong and not the question) but I would love some help understand which way the implication in the question goes and why each solution answers it being either necessary or sufficient.

Many thanks for helping me understand!