Question

1. select all the solutions, real or complex, to ((x - 2)^2 = -16).

a. (x = 6)
b. (x = -2)
c. (x = -6)
d. (x = 2 + 4i)
e. (x = 2 + 2i)
f. (x = 2 - 2i)
g. (x = 2 - 4i)

Explanation:

Step1: Take square root of both sides

Given \((x - 2)^2=-16\), take square root on both sides: \(x - 2=\pm\sqrt{-16}\)

Step2: Simplify the square root

We know that \(\sqrt{-16}=\sqrt{16}\times\sqrt{-1} = 4i\) (since \(\sqrt{-1}=i\))

Step3: Solve for x

Case 1: \(x - 2 = 4i\), then \(x=2 + 4i\)
Case 2: \(x - 2=-4i\), then \(x=2-4i\)

Answer:

D. \(x = 2 + 4i\), G. \(x = 2 - 4i\)