Simplify 3-4i/3+4i + 3+4i/3-4i

Simplify 3-4i/3+4i + 3+4i/3-4i

The Correct Answer and Explanation is:

To simplify the expression (\frac{3 – 4i}{3 + 4i} + \frac{3 + 4i}{3 – 4i}), we will perform the addition of the two complex fractions step by step.

Step 1: Find a Common Denominator

The denominators are (3 + 4i) and (3 – 4i). The common denominator will be the product of these two denominators:

[
(3 + 4i)(3 – 4i)
]

Using the difference of squares formula, we can simplify this:

[
= 3^2 – (4i)^2 = 9 – 16(-1) = 9 + 16 = 25
]

Step 2: Rewrite Each Fraction

Now we rewrite each fraction using the common denominator:

[
\frac{(3 – 4i)(3 – 4i)}{25} + \frac{(3 + 4i)(3 + 4i)}{25}
]

Step 3: Expand the Numerators

Expanding both numerators:

1. For ((3 – 4i)(3 – 4i)):

[
= 9 – 12i – 12i + 16i^2 = 9 – 24i – 16 = -7 – 24i
]

2. For ((3 + 4i)(3 + 4i)):

[
= 9 + 12i + 12i + 16i^2 = 9 + 24i – 16 = -7 + 24i
]

Step 4: Combine the Expanded Numerators

Now, we add these two results:

[
(-7 – 24i) + (-7 + 24i) = -7 – 7 + (-24i + 24i) = -14 + 0i = -14
]

Step 5: Combine the Whole Expression

Putting it all together, we have:

[
\frac{-14}{25}
]

Conclusion

Thus, the simplified form of the original expression (\frac{3 – 4i}{3 + 4i} + \frac{3 + 4i}{3 – 4i}) is:

[
-\frac{14}{25}
]

This result shows that the sum of the two complex fractions has yielded a real number, which illustrates how the imaginary parts canceled out during the addition, resulting in a simpler expression.