The Existence of Imaginary Numbers Quick Check

Question

5 of 55 of 5 Items
Question
Which is the solution to the equation 2(x−16)2=−8, re-expressed as a complex number in the standard form a±bi?(1 point)
Responses

16±2i
16 plus or minus 2 i

−4±2i
negative 4 plus or minus 2 i

16±2i2–√
16 plus or minus 2 i Start Root 2 End Root

4±2i2–√

Answer 1

To solve the equation \( 2(x - 16)^2 = -8 \), let's start by simplifying it:

1. Divide both sides by 2: \[ (x - 16)^2 = -4 \]

2. Next, take the square root of both sides, remembering that the square root of a negative number will introduce \( i \) (the imaginary unit): \[ x - 16 = \pm \sqrt{-4} = \pm 2i \]

3. Now, isolate \( x \): \[ x = 16 \pm 2i \]

Thus, the solution in standard form \( a \pm bi \) is \( 16 \pm 2i \).

The correct response is:

• 16±2i
• 16 plus or minus 2 i