the question basically is that how many quadratic functions are there satisfying

and the maximum value attained by f is 18.

since 1 is a root of , we can write

; p being the other root.

given that , we have .

also maximum value of is 18 and we know that the maxima occurs at as p and 1 are the roots of f, we have

, which simplifies to,

. now multiply both sides by k to get,

. we already have the value of which is 10, so we have,

which gives two values of k which are . both are acceptable as k had to be negative( if k is not negative then maxima of f does not exist).

corresponding values of p are -5 and -0.2.

so we have two such quadratic functions which can be found out by putting the values of k and p in