8-i-3-2i-If-the-expression-above-is-rewritten-in-the-form-a-bi-where-a-and-b-are-real-numbers-what-is-the-value-of-a-A-2-B-8-3-C-3-D-11-3

Question Number 128083 by AgnibhoMukhopadhyay last updated on 04/Jan/21

\frac{8 - i}{3 - 2 i}

If the expression above is rewritten

in the form a + b i, where a and b are

real numbers, what is the value of a ?

A . 2

B . \frac{8}{3}

C . 3

D . \frac{11}{3}

Answered by MJS_new last updated on 04/Jan/21

\frac{α + β i}{γ + δ i} = \frac{(α + β i) (γ - δ i)}{(γ + δ i) (γ - δ i)} = \frac{α γ + β δ}{γ^{2} + δ^{2}} + \frac{β γ - α δ}{γ^{2} + δ^{2}} i

Answered by A8;15: last updated on 04/Jan/21

\frac{8 - i}{3 - 2 i} = \frac{(8 - i) \cdot (3 + 2 i)}{(3 - 2 i) \cdot (3 + 2 i)} = \frac{24 - 3 i + 16 i - {2 i}^{2}}{3^{2} - {(2 i)}^{2}} =

= \frac{13 i + 24 - 2 \cdot {(\sqrt{- 1})}^{2}}{9 - 4 \cdot {(\sqrt{- 1})}^{2}} = \frac{13 i + 24 - 2 \cdot (- 1)}{9 - 4 \cdot (- 1)} =

= \frac{13 i + 26}{13} = \frac{13 i}{13} + \frac{26}{13} = 2 + i

a + bi = 2 + i

{\begin{cases} a = 2 \\ b = 1 \end{cases}

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