Question

Write the quotient 13/5 - i in the form a + bi .

Answer 1

Given Quotient:-

$\red{➤}\:$$\sf \dfrac{13}{5-i}$

To Do:-

$\orange{☛}\:$$\sf Write \: given \: Quotient \:in \: a+bi \:form$

Solution:-

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{13}{5-i} \\\end{gathered}$

Multiplaying denominator and numerator by 5+i to get denominator as real number

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{13(5+i)}{5-i(5+i)} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{5^2-i^2} \\\end{gathered}$

$\quad\quad\quad\quad\quad\sf ((a+b)(a-b)=a^2-b^2)$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{25-(-1)} \\\end{gathered}$

$\quad\quad\quad\quad\quad\sf (For\:Complex\: Number,i ; i^2 = -1)$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{25+1} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{26} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{\cancel{65}}{\cancel{26}}+\frac{\cancel{13}i}{\cancel{26}} \\\end{gathered}$

$\begin{gathered}\\\quad\longrightarrow\quad\boxed{\sf {\frac{5}{2} +\frac{1}{2} i}}\\\end{gathered}$