 I'm Zor. Welcome to Unizor education. This is problem number four in the chapter of complex numbers. We will talk right now about solving a general quadratic equation in complex numbers. And the problem is that we have to prove that we have two roots for this particular equation, and these two roots are either both real numbers or complex with opposite sign in the imaginary part. So if you have x1 and x2 as two roots of this general equation, then either both real or x1 and 2 is equal to something like a plus minus b pi, where a and b are real numbers. So both roots have the same real part, and the imaginary part is opposite sign. So that's what we have to prove actually here. And the way how we do it, we will just try to solve general equation and derive with the formula. In this case, we will derive the full formula for this type of equation and see what happens. Let's assume that all our coefficients a, b and c are real numbers. a is not equal to zero. So the roots of this equation are exactly the same as if I will divide it by a, I will have something like x squared plus px plus q equal to zero. This is actually easier. The roots are exactly the same. I just divided by a all components of this. So a is not equal to zero because it's quadratic equation. So instead of solving this, I will solve this where p is equal to p over a and q is equal to c over a. Alright, so we will solve this equation and how do we do it? I really do not remember the formula. So how can I solve quadratic equation? Well, most likely I will just try to solve something simple. For instance, if I have an equation of x square equals to whatever m, for instance, I know that the solutions are x equals plus or minus square root of m, right? A little more complicated case of this type. Just you see this px, it's kind of makes the whole thing a little bit more complex. But I will try to reduce it to this type of equation. An equation of type y square plus something like g equals to zero. So I will try to reduce this one to this one and this I know how to solve. Now, how can I do it? Well, obviously, I need something for y to combine these two together and that will be just one variable rather than kind of a polynomial expression. How can I do it? Very simply, you understand that if I will do x plus, let's say, which letter should I choose? Let's say k, for instance, square. What is this? This is x square plus 2k x plus k square, right? So I would like this to be this and I would like this to be this. So what is my k? k is obviously p divided by 2. So what I will do is the following. I will use this to have an x square plus 2k over 2x plus p square divided by 4. And this will be my y square. Now, to get back to this equation, so x is the same, p times 2 divided by 2 is the same and only this number is actually not the same. So to bring it to the following, I have to subtract p square over 4 and then add q and that's equal to 0. So this particular equation is exactly the same as this one, but now this represents a full square, which I have just symbolically used y square for this type of thing. So in this case, the equation is I can use y or I can use x plus p divided by 2. So y is in this case x plus p divided by 2. And that's why y square will be x square plus 2p divided by 2x plus p square over 4. So the equation actually becomes y square minus p square over 4 plus q is equal to 0, where y is this expression. And obviously the solution to this is y is equal to class minus square root of p minus q, from which we actually derive the x, x is equal to minus p over 2 plus minus square root of p square over 4 minus q. So this represents a general solution to our equation. And considering our coefficients are all real, we basically see that if p square over 4 minus q is positive, then both roots are real numbers, because the square root is a real number from the positive number. If, however, we have a negative number here, then the square root of this negative number is i with some kind of a real coefficient. For instance, square root of minus 4, for instance, is 2i. So basically, this represents, in this case, when this is negative, represents in the major part. So minus p divided by 2 is a real part. And plus or minus, you will have i with a certain number, a certain real number as a coefficient. So basically, you can say that this is, this represents two complex numbers with imaginary part opposite sides, opposite sides. And that's exactly what this particular problem required. And incidentally, what we have actually done, we have derived the formula for this type of equation. Or if you wish, we can return back to original equation. And instead of p and q, we can use these expressions, which wouldn't really change much. But then it will be expressed in terms of a, b, and c, the same formula, which doesn't really change the fact that either when this is positive, both roots are real numbers. Or if under the square root, I have a negative number, then I have two complex numbers with imaginary parts having opposite sides, which is exactly what was required. So thank you very much.