To simplify the expression [tex]\( \frac{4}{2 - 5i} \)[/tex], we need to eliminate the imaginary number in the denominator. This can be done by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of [tex]\(2 - 5i\)[/tex] is [tex]\(2 + 5i\)[/tex].
1. Write down the problem:
[tex]\[
\frac{4}{2 - 5i}
\][/tex]
2. Identify the complex conjugate of the denominator:
[tex]\[
\text{The complex conjugate of } (2 - 5i) \text{ is } (2 + 5i)
\][/tex]
3. Multiply both the numerator and the denominator by the complex conjugate:
[tex]\[
\frac{4}{2 - 5i} \times \frac{2 + 5i}{2 + 5i}
\][/tex]
4. Distribute in both the numerator and the denominator:
- Numerator:
[tex]\[
4 \times (2 + 5i) = 4 \times 2 + 4 \times 5i = 8 + 20i
\][/tex]
- Denominator:
[tex]\[
(2 - 5i) \times (2 + 5i) = 2^2 + 2 \times 5i - 5i \times 2 - (5i)^2
\][/tex]
[tex]\[
= 4 + 10i - 10i - 25i^2
\][/tex]
Since [tex]\(i^2 = -1\)[/tex], this simplifies to:
[tex]\[
= 4 + 0 - 25(-1) = 4 + 25 = 29
\][/tex]
5. Combine the new numerator and denominator:
[tex]\[
\frac{8 + 20i}{29}
\][/tex]
This fraction is now simplified. We can express it as separate real and imaginary parts:
[tex]\[
= \frac{8}{29} + \frac{20i}{29}
\][/tex]
Thus, the simplified form of [tex]\( \frac{4}{2 - 5i} \)[/tex] is:
[tex]\[
\frac{8}{29} + \frac{20i}{29}
\][/tex]
We can state the result in a single fraction as:
[tex]\[
\frac{8 + 20i}{29}
\][/tex]
In summary, the simplified expression is:
[tex]\[
\boxed{\frac{8 + 20i}{29}}
\][/tex]
