## Algebra: A Combined Approach (4th Edition)

The product of a complex number and its conjugate is a real number because the opposite signed imaginary parts cancel each other out (in the outer and inner steps of FOIL). In addition, when the two imaginary parts are multiplied by each other (in the last step of foil), they become $i^2$, which is equal to $-1$.
$(a+bi)\cdot(a-bi) = a^2+[abi-abi]-b^2\cdot(i^2) \\ = a^2 -b^2\cdot(-1) = a^2+b^2$ The final value does not have $i$ as any coefficient.