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​.