Simplify the expression (9 + 2i)(9 - 2i). Write your answer as a complex number.

Step-by-step solution

1. 1

   Recognise the pair as complex conjugates. The two factors differ only in the sign of the imaginary part, so they are conjugates: $z = 9 + 2i$ and $\bar z = 9 - 2i$. That pattern is worth spotting, because the answer is guaranteed to come out real and there is a shortcut.

2. 2

   Use the difference of squares. Since $(a+b)(a-b) = a^2 - b^2$ with $a = 9$ and $b = 2i$:

   $(9+2i)(9-2i) = 9^2 - (2i)^2 = 81 - 4i^2$

3. 3

   Apply $i^2 = -1$. This is the step that makes the sign flip, and skipping it produces the wrong answer $77$:

   $81 - 4(-1) = 81 + 4 = 85$

4. 4

   Verify by expanding term by term. FOIL the product without the shortcut:

   $81 - 18i + 18i - 4i^2 = 81 + 0i + 4 = 85$

   The middle terms $-18i$ and $+18i$ cancel — that cancellation is exactly why a conjugate product is always real.

5. 5

   Read off the answer as a complex number. $85 = 85 + 0i$: the imaginary part is zero. In general $z\bar z = |z|^2$, and here $|z|^2 = 9^2 + 2^2 = 81 + 4 = 85$ ✓, a third independent confirmation.