# How do you multiply (a-bi)(a+bi)?

Sep 16, 2016

$\left(a - b i\right) \left(a + b i\right) - {a}^{2} + {b}^{2}$

#### Explanation:

$\left(a - b i\right) \left(a + b i\right)$ is the product of two complex conjugate numbers and their product is always real. Such numbers always have equal real part and their imaginary part are equal in magnitude, but have opposite in sign.

While multiplying two complex numbers one should always remember that ${i}^{2} = - 1$. Using this

$\left(a - b i\right) \left(a + b i\right)$

= $a \left(a + b i\right) - b i \left(a + b i\right)$

= $a \times a + a \times b i - b i \times a - b i \times b i$

= ${a}^{2} + a b i - a b i - {b}^{2} \times {i}^{2}$

= ${a}^{2} - {b}^{2} \times \left(- 1\right)$

= ${a}^{2} + {b}^{2}$