Problem:
Quadratic polynomials P(x) and Q(x) have leading coefficients 2 and −2 , respectively. The graphs of both polynomials pass through the two points (16,54) and (20,53). Find P(0)+Q(0).
Solution:
Answer (116):
Because the leading coefficients of P(x) and Q(x) are negatives of each other, the polynomial R(x)=P(x)+Q(x) is linear. Furthermore, R(16)=54+54=108 and R(20)=53+53=106. It follows that R(x)=116−0.5x, so P(0)+Q(0)=R(0)=116. Note that
P(x)=2x2−4289x+698 and Q(x)=−2x2+4287x−582.
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions