2018 AMC 10A Problem 10

Name the two radicals: $u = \sqrt{49 - x^2}$ and $v = \sqrt{25 - x^2}$. We are told $u - v = 3$ and asked for $u + v$.

Squaring each radical eliminates $x$ from their difference:
$$u^2 - v^2 = (49 - x^2) - (25 - x^2) = 24.$$

By the difference of squares, $u^2 - v^2 = (u - v)(u + v)$, so
$$3 \cdot (u + v) = 24 \quad\Longrightarrow\quad u + v = 8.$$

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

When you are given $u - v$ and asked for $u + v$ (or the reverse), look for $u^2 - v^2$: it factors as the product of the two, and here the $x^2$ terms cancel, leaving a constant. Recognizing that the question never requires $x$ itself saves the whole algebraic slog of isolating a radical and squaring twice.

Grind it out: isolate one radical and square, $\sqrt{49-x^2} = 3 + \sqrt{25-x^2}$, so $49 - x^2 = 9 + 6\sqrt{25-x^2} + 25 - x^2$, giving $\sqrt{25 - x^2} = \tfrac{15}{6} = \tfrac52$ and then $\sqrt{49-x^2} = \tfrac{11}{2}$; their sum is $8$. It works, but it is three times as much writing.

• Solving for x by squaring the given equation, which is slow and error-prone, when the sum can be found without ever knowing x.
• Computing $u^2 - v^2 = 49 - 25 = 24$ correctly but then dividing $24$ by $u + v$ instead of by $u - v$, or forgetting the factorization and guessing (C) or (E).