2023 AMC 12B Problem 12

Problem 12 of 25IntermediateAlgebra

For complex numbers

u=a+bi

and

v=c+di,

define the binary operation

\otimes

u\otimes v=ac+bdi.

Suppose

z

is a complex number such that

z\otimes z=z^2+40.

What is

|z|?

Show solution

Solution

With

z=a+bi,

we have

z\otimes z=a^2+b^2 i

and

z^2+40=(a^2-b^2+40)+2abi.

The real parts give

a^2=a^2-b^2+40,

b^2=40.

The imaginary parts give

b^2=2ab,

b=2a

and

a^2=\tfrac{b^2}{4}=10.

Then

|z|^2=a^2+b^2=50,

|z|=5\sqrt{2}.

Thus, the correct answer is E.