If the product of the complex numbers 4 - 2i and k + 6i is a real number, where k is a rea… — ACT

If the product of the complex numbers $4 - 2i$ and $k + 6i$ is a real number, where $k$ is a real constant and $i = \sqrt{-1}$ , what is the value of $k$ ?

Answer: 12

Answer

The product of the complex numbers

(4 - 2i)(k + 6i)

expands to

4k + 24i - 2ki - 12i^2

. Substituting

i^2 = -1

simplifies the expression to

(4k + 12) + (24 - 2k)i

. For this expression to represent a real number, the imaginary part must be zero:

24 - 2k = 0

, which yields

k = 12

Step-by-Step Solution

Multiply the two complex numbers

(4 - 2i)

and

(k + 6i)

using the FOIL method.

4k + 24i - 2ki - 12i^2

To find the product of the two complex expressions.

Substitute

i^2 = -1

into the expression and group the real and imaginary parts.

(4k + 12) + (24 - 2k)i

To simplify the expression into standard complex form

a + bi

Set the imaginary part of the resulting complex number to

0

and solve for

k

24 - 2k = 0 \implies k = 12

A complex number is real if and only if its imaginary part is equal to zero.

Key Concept

Complex multiplication and the definition of a real number in the complex plane

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