Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$. The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has slope $2$. Find $a + b$.
Using the slope formula gives \(\dfrac{b^2 - a^2}{b - a} = 2\).
Note that \(b^2 - a^2 = (b - a)(a + b)\) by difference of squares formula. Then, simplifying, we have
\(\dfrac{(b - a)(a + b)}{b - a} = 2\\ a + b = \boxed{2}\)
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