2018 AMC 10A Problem 10

Give the two square roots short names so the pattern is easier to see:
$$u = \sqrt{49 - x^2}, \qquad v = \sqrt{25 - x^2}.$$
We are told $u - v = 3$ and asked for $u + v$.

1. Square each root. Squaring undoes a square root: $u^2 = 49 - x^2$ and $v^2 = 25 - x^2$.
2. Subtract. The $x^2$ parts cancel:
$$u^2 - v^2 = (49 - x^2) - (25 - x^2) = 24.$$
So even without knowing $x$, we know $u^2 - v^2 = 24$.
3. Use difference of squares. For any two numbers, $u^2 - v^2 = (u - v)(u + v)$. Put in what we know:
$$3 \cdot (u + v) = 24.$$
4. Solve. $u + v = 8$.

The answer is $\boxed{\textbf{(A)}\ 8}$.

When a problem gives you "big minus small" and asks for "big plus small", think of the difference of squares: the two multiply to $u^2 - v^2$. Here that product is easy because the $x^2$ parts cancel, leaving a plain number. Before solving for a variable, check whether the question actually needs it.

Work out each square root on its own. Move one root across: $u = 3 + v$. Square both sides: $49 - x^2 = 9 + 6v + 25 - x^2$, so $15 = 6v$ and $v = \tfrac52$. Then $u = 3 + \tfrac52 = \tfrac{11}{2}$, and $u + v = \tfrac{11}{2} + \tfrac52 = 8$. This works too, but it takes more steps.