Problem:
A quadratic equation ax2−2ax+b=0 has two real solutions. What is the average of the solutions?
Answer Choices:
A. 1
B. 2
C. ab
D. a2b
E. 2a−b
Solution:
The quadratic formula implies that the two solutions are
x1=2a2a+4a2−4ab and x2=2a2a−4a2−4ab
so the average is
21(x1+x2)=21(2a2a+2a2a)=(A)1
OR
The sum of the solutions of a quadratic equation is the negative of the coefficient of the linear term divided by the coefficient of the quadratic term. In this case the sum of the solution is a−(−2a)=2. Hence the average of the solutions is (A)1.