 I'm Zor. Welcome to Unisor Education. I will solve the problem number three for complex numbers right now. And this is an equation which we would like to solve. First of all, let me just say upfront that it's quite obvious that this equation cannot be solved in real numbers because obviously it can be changed to x to the second plus 4x plus 4 plus 1 equal to 0. And this represents the full square, x plus 2 square plus 1 equal to 0. So obviously, this is a positive number and it cannot be equal to 0 if you add 1. So we do need complex numbers to solve certain quadratic equations. But in this particular case, I will just use whatever simplification I just made to solve this equation quite easily. I don't need the formula basically to solve this equation because I can immediately write from here that x plus 2 square is equal to minus 1 from which we derive immediately that x plus 2 is equal to plus or minus i. Where i is a complex number, which square, which is equal to minus 1. Why is it plus or minus i in front of i? Well, obviously because i square is minus 1 and minus i square is also minus 1. So whenever we are extracting the square root from any number, we should really not forget plus and minus in this particular case. Well, and obviously from here the solution is x is equal to minus 2 plus or minus i. We have two solutions, which is exactly the right thing for quadratic equation. Maybe a little later we will talk about that any equation of nth degree in complex numbers has exactly n roots. Some of the roots can be multiple roots, so to speak. But anyways, n roots in this case, quadratic equation, the power is 2, so we have two different roots for this equation. What's next step? Never forget to check your answer. Let's just check it out. Okay, x is equal to minus 2 plus or minus i. First of all, x square minus 2 plus i square. Let's check for plus first. Plus 4x plus 4 minus 2 plus i plus 5 equals... ...minus 2 square is 4 minus 4i plus i square, right? ...minus 8 plus 4i plus 5 equals... ...minus 4i and plus 4i are reduced. 4 minus 8 is minus 4 plus 5 is 1 plus i square, which is 1 minus 1, because i square is minus 1, which is equal to 0. Exactly right. So that's the verification of the first root. And now let's do the second one. It should be very similar. So it's minus 2 minus i square plus 4 times minus 2 minus i plus 5 equals 4 plus 4i plus i square. That's this guy. ...minus 8 minus 4i plus 5 equals... ...4i minus 4i is 0. 4 minus 8 minus 4 plus 5 is 1 plus i square, 1 minus 1 is 0. Both roots fit. Check is done. The equation is solved. By the way, the way how I solved this equation wasn't really to apply any kind of formula. And I do prefer to derive certain results without actually remembering any formula. I would rather do it the way how I just derived, how I just solved this equation by just noticing certain things, the full square and something like this. For equations of this type of power 2, that's probably the best if you don't remember the formula. The formula is rather long, so you might forget it. That's okay. As long as you don't forget how to derive this formula, it's okay to forget it. Thanks very much. Hope you enjoyed it.