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 $(a,b).$
This is a quadratic equation, so I know that the sum of its solutions is equal to the opposite of the coefficient of $x$, and the product of its solutions is equal to the constant term. That means that $a+b=-a$ and $ab=b$. I can simplify these equations by adding $a$ to both sides of the first one and dividing by $b$ on both sides of the second one. This gives me $2a=-b$ and $a=\frac{b}{b}=1$. Since $a=1$, I can substitute it into the equation $2a=-b$ and get $b=-2$. So the ordered pair $(a,b)$ is $(1,-2)$. 