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.
Lily made an error. Explain Lily's error and correct the step which contains the error.

Answer 1

step 1 says
\[ (2+i)^2 = 4+i^2\]
that is wrong.

Answer 2

\[ (2+i)^2 \ \text{ means } (2+i)(2+i) \]
I would multiply that out the same way you would multiply for example (a+b)(a+b)

Answer 3

so that part just cancels itself out?

Answer 4

do you know how to multiply (x+y)(x+y) ?
some people use FOIL
or use the distributive property
for example, let A be the first (x+y).
we (should!) know A(x+y) can we written as A*x + A*y
but A is short for (x+y) so that is (x+y)x + (x+y)y
now distribute again: x*x + xy + xy + y^2
or
x^2 +2xy + y^2

we use that same idea on (2+i)(2+i)

Answer 5

okay so the first step would then look like (2+i)^2 = 2^2+i^2+4i

Answer 6

yes, exactly

next, use the simplification that i*i is -1 (by definition)
so you get
\[ (2+i)^2 = 2^2+i^2+4i \\
= 4-1+4i \\
= 3+4i \]

Answer 7

okay so is that all the step redone or is there more that needs to be changed?

Answer 8

the rest of the steps up above are ok, but don't make sense because they started wrong.
I would say:
step 1: (2+i)(2+i) = 4 + 4i+i^2
step 2: i^2 =1 so 4 + 4i-1
step 3: 3+4i

Answer 9

okay thank you so much!