Question

What is the standard form of (2-4i)(3+5i) divided by (3+i)

Answer 1

First we need to simplify the numerator of the fraction

(2 - 4i)(3 + 5i)
= 6 +10i - 12i - 20i²
= 6 - 2i + 20
= 26 - 2i

So now the fraction becomes:

$\frac{26-2i}{3+i}$

In order to convert the fraction to standard form, we need to multiply and divide the fraction by the conjugate of the denominator as shown below:

$\frac{26-2i}{3+i}* \frac{3-i}{3-i} \\ \\ = \frac{78-26i-6i+2i^{2} }{9-i^{2} } \\ \\ = \frac{78-32i-2}{9+1} \\ \\ = \frac{76-32i}{10} \\ \\ = \frac{76}{10}- \frac{32i}{10} \\ \\ = \frac{38}{5}- \frac{16}{5}i$