Question

I need help doing this never got taught this before

Answer 1

`(-14)/(25)\text{ (option C)}`

Step-by-step explanation:

`\frac{3-4i}{3\text{ + 4i}}\text{ +}\frac{3+4i}{3\text{ - 4i}}`

Simplifying:

`\begin{gathered} =\frac{(3-4i)(3\text{ - 4i) + (3+ 4i)(3 + 4i)}}{(3\text{ + 4i)(3 - 4i)}} \\ =\frac{3(3\text{ - 4i)-4i(3 - 4i)}+\text{ \lbrack}3(3\text{ + 4i)+4i(3 + 4i)\rbrack}}{(3\text{ + 4i)(3 - 4i)}} \end{gathered}`
`\begin{gathered} =\text{ }\frac{9-12i-12i+16i^2+\text{ \lbrack}9+12i+12i+16i^2\text{\rbrack}}{(3\text{ + 4i)(3 - 4i)}} \\ =\text{ }\frac{9-12i-12i+16i^2+\text{ \lbrack}9+12i+12i+16i^2\text{\rbrack}}{3\text{ (3 - 4i)+ 4i (3 - 4i)}} \\ =\text{ }\frac{9-12i-12i+16i^2+\text{ \lbrack}9+12i+12i+16i^2\text{\rbrack}}{9\text{ - 12i+12i - 1}6i^2} \end{gathered}`
`\begin{gathered} =\text{ }\frac{9-24i+16i^2+\text{ }9+24i+16i^2}{9\text{ - 12i+12i - 1}6i^2} \\ =\text{ }\frac{9-24i+24i+\text{ }9+16i^2+16i^2}{9\text{ - 12i+12i - 1}6i^2} \\ =\text{ }\frac{9+\text{ }9+16i^2+16i^2}{9\text{ - 1}6i^2} \\ =\text{ }\frac{18+32i^2}{9\text{ - 1}6i^2} \end{gathered}`

`=(-14)/(25)\text{ (option C)}`