2013 AMC 10B Problems/Problem 19
Problem
The real numbers form an arithmetic sequence with The quadratic has exactly one root. What is this root?
I didn't really understand this problem...
If there is one root, the discriminant is 0. Therefore, . Let's define and . If , by substitution, and distributing, . Thus, through simple arithmetic, and . Then, the root, by quadratic formula, is $\frac{-b+\sqrt{b^2-4ac}}\{2a}.$ (Error compiling LaTeX. Unknown error_msg) Since the discriminant is assumed as zero, the root will be , or or . This is equal to $-b/2b-\sqrt{3}\b$ (Error compiling LaTeX. Unknown error_msg) or . Multiplying by the conjugate gives .