Problem:
The real numbers c,b,a form an arithmetic sequence with a≥b≥c≥0. The quadratic ax2+bx+c has exactly one root. What is this root?
Answer Choices:
A. −7−43
B. −2−3
C. −1
D. −2+3
E. −7+43
Solution:
Let the common difference in the arithmetic sequence be d, so that a=b+d and c=b−d. Because the quadratic has exactly one root, b2−4ac=0. Substitution gives b2=4(b+d)(b−d), and therefore 3b2=4d2. Because b≥0 and d≥0, it follows that 3b=2d. Thus the real root is