Consider the quadratic function. f(x) = px (q-x), where p and q are positive integers. the graph of y = f(x) passes through the point (6,0). the vertex of the function is (3, 27) find the value of p.

p=3 and q = 6

Step-by-step explanation:

f(x) = px (q-x) = pxq -x²p

f(x) = -px² + pqx

you have the system : f(6) = 0........(*)

                                    f(3) = 27.....(**)

(*) means : when x =6    y = 0    : -p(6)² + pq(6) = 0    .... -36p +6pq =0

(**) means : when x =3   y = 27 : -p(3)² +pq(3) = 27........ -9p +3pq = 27

so the system : -6p +pq =0...(1)

                         -3p +pq = 9 ...(2)

by (1) : pq = 6p

by(2) : pq = 3p +9

so : 6p =3p+9

3p = 9

p=3  put this value in (1) : -6(3)+3q =0

3q =18

q=6    so : f(x) = -3x²+18x

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