Suppose that real number x satisfies √49-x^2-√25-x^2=3. What is the value of √49-x^2+√25-x^2?

2018 AMC 10A Problem 10 — Solution

Attempt Problem 10 of the 2018 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

10.

Suppose that real number $x$ satisfies $\sqrt{49-x^2}-\sqrt{25-x^2}=3.$ What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}?$

$8$

$\sqrt{33}+8$

$9$

$2\sqrt{10}+4$

$12$

Note that the left hand side of the equation and the desired expression are conjugates. Multiplying them would remove the square roots.

Multiplying them yields $49 - x^2 - 25 + x^2 = 24.$

This means that the product of the values of the expressions is equal to $24.$ The desired value is therefore $24 \div 3 = 8.$

Thus, A is the correct answer.