Under what conditions is √a^2 + b^2 = a + b true, where a and b are real numbers?

(D) It is true if and only if ab = 0 and a + b ≥ 0.

(B) It is true if and only if ab = 0.

(C) It is true if and only if a + b ≥ 0.

2021 AMC 12A Spring Problem 2 — Solution

2021 AMC 12A Spring Problem 2

Below is the professionally curated solution for Problem 2 of the 2021 AMC 12A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Spring solutions, or check the answer key.

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Concepts:radical algebraic manipulation

Difficulty rating: 1200

Under what conditions is $\sqrt{a^2 + b^2} = a + b$ true, where $a$ and $b$ are real numbers?

It is never true.

It is true if and only if $ab = 0.$

It is true if and only if $a + b \ge 0.$

It is true if and only if $ab = 0$ and $a + b \ge 0.$

It is always true.

Solution:

Because $\sqrt{a^2+b^2}$ is never negative, equality requires $a + b \ge 0.$ Squaring both sides gives $a^2 + b^2 = (a+b)^2 = a^2 + 2ab + b^2,$ which simplifies to $2ab = 0,$ i.e. $ab = 0.$

Conversely, if $ab = 0$ then $a^2 + b^2 = (a+b)^2,$ and if additionally $a + b \ge 0$ then $\sqrt{a^2+b^2} = |a+b| = a+b.$ So both conditions together are exactly what is needed.

Thus, the correct answer is D.

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