Question

express 2+i/2-i in the form a+ib

Answer 1

ANSWER:

Given:

• 2+i / 2-i

To Do:

• Convert in a + ib form.

Solution:

$\text{We are given that,}\\\\:\longrightarrow\dfrac{2+i}{2-i}\\\\\text{On rationalising,}\\\\:\implies\dfrac{2+i}{2-i}\\\\:\implies\dfrac{2+i}{2-i}\times\dfrac{2+i}{2+i}\\\\:\implies\dfrac{(2+i)^2}{(2-i)(2+i)}\\\\\text{We know that,}\\\\:\hookrightarrow(a+b)(a-b)=a^2-b^2\\\\\text{And,}\\\\:\hookrightarrow(a+b)^2=a^2+2ab+b^2\\\\\text{So,}\\\\:\implies\dfrac{(2+i)^2}{(2-i)(2+i)}\\\\:\implies\dfrac{2^2+2(2)(i)+i^2}{2^2-i^2}\\\\\text{We know that,}\\\\:\hookrightarrow i^2=-1\\\\:\implies\dfrac{4+4i-1}{4-(-1)}$

$:\implies\dfrac{4-1+4i}{4+1}\\\\:\implies\dfrac{3+4i}{5}\\\\\bf{:\implies\dfrac{3}{5}+\dfrac{4i}{5}}$

Formulae Used:

• (a + b)(a - b) = a² - b²
• (a + b)² = a² + 2ab + b²
• i² = -1