Find that Root!

This problem was adapted from AMC 10B 2013, Problem 19.

The real numbers $c,b,a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$ . The quadratic equation $ax^{2}+bx+c$ has exactly one root. If this root can be expressed in the form $-b+\sqrt{a}$ , what is $a+b$ ?

[This question is not clearly phrased.]