Compute the following and write in the form x+iy :

[tex]\frac{1+2i}{3-4i} + \frac{2-i}{5i}[/tex]

[tex] \Large{\boxed{\sf \dfrac{1 + 2i}{3 - 4i} + \dfrac{2 - i}{5i } = - \dfrac{2}{5}}} [/tex]

[tex] \\ [/tex]

Given sum:

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i}[/tex]

[tex] \\ [/tex]

We can simplify the sum only if the denominators of the two fractions are the same. Since they are different, we have to multiply the numerator and the denominator of each fraction by the denominator of the other one.

[tex] \sf \: \dfrac{1+2i}{3-4i} + \dfrac{2-i}{5i} = \dfrac{5i(1 + 2i)}{5i(3 - 4i)} + \dfrac{(3 - 4i)(2 - i)}{(3 - 4i)5i} \\ \\ \\ \sf \: = \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } [/tex]

[tex] \\ [/tex]

Replace i² with -1:

[tex] \sf \dfrac{5i + 10 {i}^{2} }{15i - 20 {i}^{2} } + \dfrac{6 - 3i - 8i + 4 {i}^{2} }{15i - 20 {i}^{2} } \: \\ \\ \\ \\ \sf \: = \dfrac{5i + 10( - 1)}{15i - 20( - 1)} + \dfrac{ 6 - 11i + 4( - 1)}{15i - 20( - 1)} \\ \\ \\ \\ \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} [/tex]

[tex] \\ [/tex]

Simplify the expression:

[tex] \sf = \dfrac{5i - 10}{20 + 15i} + \dfrac{2 - 11i}{20 + 15i} \\ \\ \\ \\ \sf \: = \dfrac{5i - 10 + 2 - 11i}{ 20 + 15i} = \sf \dfrac{ - 8 - 6i}{20 + 15i}[/tex]

[tex] \\ [/tex]

To write our solution in the x + iy form, also known as the algebraic form, we have to understand what the conjugate of a complex number is.

[tex] \textsf{Let z be our complex number, and} \: \overline{\sf z} \: \textsf{its conjugate.} [/tex]

[tex] \\ [/tex]

The conjugate of z, [tex] \overline{ \sf z}, [/tex] is the complex number formed of the same real part as z but of the opposite imaginary part.

Since x is the real part of z, and y is its imaginary part, this can be expressed as:

[tex] \sf If \: z = x + iy \:, then \: \overline{ \sf z} = x - iy [/tex]

[tex] \\ [/tex]

Now, we have to multiple both the denominator and the numerator of our fraction by the conjugate of its denominator:

[tex]\sf \dfrac{ - 8 - 6i}{20 + 15i} = \dfrac{( - 8 - 6i)( \overbrace{20 - 15i}^{ \overline{z}}) }{ (20 + 15i)( \underbrace{20 - 15i}_{ \overline{z}}) } \\ \\ \\ \sf = \dfrac{ - 160 + 120i - 120i + 90 {i}^{2} }{400 - 300i + 300i - 225 {i}^{2} } \\ \\ \\ \sf = \dfrac{ - 160 + 90 {i}^{2} }{400 - 225 {i}^{2} }[/tex]

[tex] \\ [/tex]

One more time, substitute -1 for i²:

[tex] \sf \: \dfrac{ - 160+ 90 {i}^{2} }{400 - 225 {i}^{2} } \: = \dfrac{ - 160 + 90( - 1)}{400 - 225( - 1)} \\ \\ \\ \sf = \boxed{\sf - \dfrac{ 250}{625}} [/tex]

[tex] \\ [/tex]

Finally, let's simplify our result:

[tex] \sf - \dfrac{250}{625} = - \dfrac{2 \times 125}{5 \times 125} = \boxed{ \boxed{ \sf - \dfrac{2}{5}}}[/tex]

[tex] \\ \\ [/tex]

▪️Learn more about the algebraic form of a complex number and the conjugate of a complex number here:

↣https://brainly.com/question/25633236

Which type of graph(s) should be used for the following data? Choose all that apply. (Check photo)

(please answer if you know) On a​ map, 1 inch equals 5.2 miles. Two houses are 3.5 inches apart on the map. What is the actual distance between the​ houses? Use pencil and paper. Show how you can represent the scale with two different ratios. What ratio is more helpful for solving the​ problem? Explain.

Problem Suppose an archaeologist finds a camel tooth that contains 42 % of original amount of C-14. Find age of the tooth. N = Noekt No = initial mass of C-14 (at time t= 0) N = amount of C-14 at time t k= 0.0001 t = time, in years Since, our equation can be written like this: 0.42 No = Noe Noe -0.00010​

izllllllllllllllllllll

The area of a rectangle with a
length of 10 cm and a width of 3
cm.

A box of donuts cost $9. You want to send donuts to the local nursing home. Set up an equation to find how many boxes you can send if you have $72.

Please help.. You can answer one if you want but whoever answers both get brainliest

giving out brainliest answer !!!! help me out asap please !!!!

What is the length of missing side Y Options are on photo

The sales rate is 8%. If Janelle buys a gymnastics mat priced at $87 , how much tax will she pay?

-7x + 2y = -5
y = -7x + 8
explain how please

Jill wants to paint the front and top surfaces of each step leading to her porch. There are 6 steps like the one shown. How many total square inches of the steps will she paint?

A: 1,296

B: 1,776

C: 3,552

D: 5,760​

Hey guys, I just got braces and they are really uncomfy so how long did it take yall to get used to them?

what is the slope of 4x-3y=18

Find the distance between the points (4.5,-1.5), (4.5,7.25)
The distance is ____.

What is the volume if
the base is 48 square inches and height is 8 inches

PLEASE HELP !! ILL GIVE BRAINLIEST !! 100 POINTS

WILL MARK BRAINLY FOR WHO EVER GETS THIS RIGHT

I REALLLY NEEED THISSSS ANSWERRRR PLZZZZ I'M GOING TO FAILLL I ONLY NEEEED ONE MOREEEE.
Give an example of other data that can
be represented using a histogram.
________________________________________
________________________________________

PLs help me. I need to find the measure of B

what is -3/4 +7/2 in simplest form

At a school carnival, 51 out of 68 tickets sold were early-admission tickets. What percentage of the tickets were early-admission tickets?

please help i have been stuck on this for 3 hours

⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️⚠️HELP HELP HELP HELP - A triangle has side lengths of 7, 23, and 25. Is the triangle a right triangle? *
Yes, the triangle is a right triangle.
No, the triangle is not a right triangle.
There is not enough information to tell if the triangle is a right triangle.

if 40% percent is a number of 35 what is 50% of that number

Anyone who know what the answer please please?!

1. Tamekia and Marsha mow lawns during the summer to earn money. Tamekia
determined that she can earn between $6.00 and $6.25 per hour. Marsha
estimates that she earns between $7.50 and $8.00 per hour. About how much
more money will Marsha earn than Tamekia if they each work 22 hours? *
A. $65.01 to $85.00
B. $45.01 to $65.00
C. $25.01 to $45.00
D. $5.00 to $25.00

How many 4 digit numbers are multiples of 8?

BRAINLIEST!!! find the probability for numbers 11 and 13 please

Pleaseee helppp me on this