For the complex number z = \frac{3 + 5i}{2 - i}, where i = \sqrt{-1}, what is the real par… — ACT

For the complex number $z = \frac{3 + 5i}{2 - i}$ , where $i = \sqrt{-1}$ , what is the real part of $z$ ?

$\frac{1}{5}$ Answer
B
$\frac{11}{5}$
C
$\frac{3}{2}$
D
$\frac{1}{3}$
E
$-\frac{1}{5}$

Answer

The real part of the complex number is

\frac{1}{5}

To find the real part of the complex fraction, we multiply both the numerator and the denominator by the complex conjugate of the denominator,

2 + i

. This yields a denominator of

(2-i)(2+i) = 4 - i^2 = 5

. The numerator becomes

(3+5i)(2+i) = 6 + 3i + 10i + 5i^2 = 1 + 13i

. Writing the result in standard form gives

\frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

Step-by-Step Solution

Identify the complex conjugate of the denominator, which is

2 + i

, and set up the multiplication for both the numerator and the denominator of

z

\frac{(3 + 5i)(2 + i)}{(2 - i)(2 + i)}

Multiplying the numerator and denominator by the conjugate of the denominator allows us to simplify the expression and eliminate the imaginary unit from the denominator.

Expand the numerator using binomial multiplication and simplify using the identity

i^2 = -1

6 + 3i + 10i + 5i^2 = 6 + 13i - 5 = 1 + 13i

This determines the simplified complex expression in the numerator.

Expand the denominator and simplify using the identity

i^2 = -1

(2 - i)(2 + i) = 4 - i^2 = 4 - (-1) = 5

This simplifies the denominator into a real number.

Combine the simplified numerator and denominator to write

z

in standard form

a + bi

and identify the real part

a

z = \frac{1 + 13i}{5} = \frac{1}{5} + \frac{13}{5}i

, so the real part is

\frac{1}{5}

Expressing the complex number in standard form separates the real and imaginary components.

Key Concept

Complex Division and Simplification

Estimated Time:1m 30s

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