Question

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.

Answer: D

Concepts:radical algebraic manipulation

Difficulty rating: 1200

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.

2021 AMC 12A Spring Problem 2

Answer: D

Difficulty rating: 1200

Solution:

Problem 2 in Other Years