Question

Lily is practicing multiplying complex numbers using the complex number (2+i).

To determine the value of (2+i)²,Lily performs the following operations:
Step 1: (2+i)²= 4+i²
Step 2: 4+i² = 4+ (−1)
Step 3: 4+(-1)=3
Lily made an error.
Explain Lily's error and correct the step which contains the error.
Bonus:
Lily is continuing to explore different ways in which complex numbers can be multiplied so the
answer is not a complex number. Lily multiplies (2+i) and (a+bi), where a and b are real
numbers, and finds that her answer is not a complex number.
A. Write an equation that expresses the relationship between a and b.

Answer 1

Lily's error is in the first step, (2+i)^2 ≠ 4 + i^2.

(2+i)^2 = (2+i)(2+i) and you need to FOIL.

(2+i)^2 = (2+i)(2+i)

= 4 + 2i + 2i + i^2

= 4 + 4i + (-1)

= 3 + 4i

Bonus:

If you want to multipy (2+i) by (a+bi) and not end up with a complex number, you'd first FOIL

(2+i)(a+bi) = 2a + 2bi + ai + bi^2

We know i^2 = -1, so this becomes

= 2a + 2bi + ai + b(-1)

= 2a + 2bi + ai - b

= 2a - b + 2bi + ai

Now, for this not to be complex, we need the imaginary pieces to cancel each other out. In other words 2bi+ai=0. For that to happen, 2b + a = 0, or

2b = -a or a = -2b

So it would seem that if we pick any b-value and make a = -2b, then we'll end up with a non-complex number.

Let's try b=5, making a = -10

(2+i)(-10+5i) = -20 + 10i - 10i + 5i^2

The 10i's cancel and 5i^2 = -5, so we're left just with -25.