How to solve a problem with complex numbers (not a request to solve, but to prompt)?

Question

Mathematics

Unfortunately, the teacher did not give the necessary material, but only gave tasks, what to read and how to solve it the easiest way?

Answer 1

In the first and second tasks, it is necessary to reduce the complex number to the form x + iy, so that i is one in the expression. To do this, you need to get rid of fractions with i in the denominator.
In the third problem, you just need to solve the equation. There will be a negative discriminant, the root of which is a complex number.

Answer 2

1, 2 - bring to a common denominator, then multiply the numerator and denominator by the expression conjugate to the denominator so that the formula for the difference of squares is applicable below. This will get rid of i in the denominator.
3 - I'll write later if necessary. Offhand, substitute the roots of the equation into the equation itself and find a.

Answer 3

1,2 - first multiply the numerator and denominator of the first fraction by 1-i, and the second fraction by 5i+2. You get two fractions that have a real number in the denominator (the first fraction has 2, the other has -29). Now multiply both fractions by 58 (for this, the numerator of the first fraction is multiplied by 29, and the second by -2). It turns out the number 29*(1-i)+2*5*(5*i+2). Open the brackets and get the answer.
3 - just take the formula of the roots of the quadratic equation. Everything is visible there.

Answer 4

wolfram
Can you guess what it is?