QuestionFind the quotient and express it as $a + b i$ : $\frac{3-i}{4+3 i}$ . Simplify your answer.

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the form of a quotient $\frac{3-i}{4+3i}$ . . We need to simplify this quotient and write the result in the form $a+bi$ , where $a$ and $b$ are real numbers.

STEP 2

To simplify a complex fraction, we usually multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a+bi$ is $a-bi$ . So, the conjugate of $4+i$ is $4-i$ .
$\frac{-i}{4+i} \times \frac{4-i}{4-i}$

STEP 3

Now, multiply the numerators together and the denominators together. Remember that $i^2 = -1$ .
umerator $(3-i)(-3i) =12 -9i -i +3i^2 =12 -13i -3 =9 -13i$
Denominator $(+3i)(-3i) =16 -12i +12i -9i^2 =16 +9 =25$
So, the fraction becomes $\frac{9-13i}{25}$

STEP 4

Now, we can write the fraction in the form $a+bi$ . To do this, we divide the real part and the imaginary part by the denominator separately.
$\frac{9}{25} - \frac{13}{25}i$

STEP 5

This is the simplified form of the given complex fraction. So, the solution is $\frac{9}{25} - \frac{13}{25}i$

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