# When -3-2i is multiplied by its conjugate, what is the result?

##### 1 Answer

The result is

#### Explanation:

A binomial's **conjugate** is the same expression but with the opposite sign on the 2nd term. For example:

For any complex number **complex conjugate** is

#(a+bi)(a-bi)=a^2-(ab)i+(ab)i-b^2i^2#

#color(white)((a+bi)(a-bi))=a^2-cancel((ab)i)+cancel((ab)i)-b^2("-"1)#

#color(white)((a+bi)(a-bi))=a^2+b^2#

See what happened? The imaginary part is no longer there. Multiplying a complex number by its conjugate turns it into a completely real number.

In this example, we're asked to multiply

#("-"3-2i)("-"3+2i)=9-6i+6i-4i^2#

#color(white)(("-"3-2i)("-"3+2i))=9-cancel(6i)+cancel(6i)-4("-"1)#

#color(white)(("-"3-2i)("-"3+2i))=9+4#

#color(white)(("-"3-2i)("-"3+2i))=13#

## Bonus:

The quick trick to find the product of a complex number and its conjugate: square each coefficient, and then add the squares. That'll be your result. (That's actually right from the general formula above.)