Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$.  Find the ordered pair $(a,b).$
By Vieta's formulas, $a + b = -a$ and $ab = b.$  Since $b$ is nonzero, $a = 1.$  Then $b = -2a = -2,$ so $(a,b) = \boxed{(1,-2)}.$