If (5 - 2i) - (2 + ki) = 3 + 4i, where i = \sqrt{-1} and k is a constant, what is the valu… — ACT

If $(5 - 2i) - (2 + ki) = 3 + 4i$ , where $i = \sqrt{-1}$ and $k$ is a constant, what is the value of $k$ ?

Answer: -6

The value of the constant

k

-6

Distributing the subtraction sign across the second complex number gives

(5 - 2i) - 2 - ki = 3 + 4i

. Combining the real parts (

5 - 2 = 3

) and grouping the imaginary parts yields

3 + (-2 - k)i = 3 + 4i

. Since the real parts are equal, we set the coefficients of the imaginary parts equal to each other:

-2 - k = 4

. Solving for

k

gives

k = -6

Distribute the negative sign to the expression

(2 + ki)

-2 - ki

To remove the parentheses, the subtraction must apply to all terms inside the parentheses.

Group and combine the real parts and imaginary parts on the left side of the equation

3 - (2 + k)i

Grouping like terms allows us to express the left side as a standard complex number

a + bi

Equate the imaginary parts from both sides of the equation

-2 - k = 4

For two complex numbers to be equal, their corresponding real parts and imaginary parts must be equal.

Solve the linear equation for

k

k = -6

Isolating

k

gives the value that satisfies the original equation.

Equality of complex numbers and operations of addition/subtraction on complex numbers.

Estimated Time:45s

Question