Question

$\sf{For\:Creative\:Brainly\:Challengers}$
There are two numbers α and β. If two equations are given by $\left \{ {{\alpha +\beta =7+2\sqrt{3}i } \atop {\alpha \beta =5+6i}} \right.$, find the sum of the reciprocal of two numbers. (Quadratic equation is not allowed. & Show your steps.)

Answer 1

Given :

$\sf \alpha +\beta =7+2\sqrt{3}\ i$

$\sf \alpha \beta=5+6\ i$

To Find :

Sum of the reciprocal of two numbers

$\to \sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

Solution :

$\\ \sf \dfrac{1}{\alpha}+\dfrac{1}{\beta}$

$\\ \to \sf \dfrac{\alpha +\beta}{\alpha \beta}$

$\\ \to \sf \dfrac{7+2\sqrt{3}\ i}{5+6\ i}$

Rationalize the denominator ,

$\\ \to \sf \dfrac{7+2\sqrt{3}\ i}{5+6\ i}\times \dfrac{5-6\ i}{5-6\ i} \\\\\to \sf \dfrac{35-42\ i+10\sqrt{3}\ i+12\sqrt{3}}{5^2+6^2}\\\\\to \sf \dfrac{35-42\ i+\sqrt{3}(10\ i+12)}{61}$

Point To Be Noted :

$\bigstar\ \; \sf i^2=-1$

where ,

• i is complex number

Answer 2

Solution:-

=> 1/a + 1/b

=> a+b/ab

⠀⠀⠀=> 7+2√3i / 5+6i

Rationalize the denominator,

→ 5+6 i7+2 3 i × 5−6 i5−6 i

→ 5 2 +6 235−42 i+10 3 i+123

→ 6135−42 i+ 3 (10 i+12)

Point To Be Noted :

i² = −1

where ,

• i is complex number