## mathslover Group Title Show that every complex number is a root of some quadratic equation. one year ago one year ago

1. waterineyes Group Title

I forgot to say : Congratulations for $$\large \color{green}{\frak{green..}}$$ @mathslover

2. mathslover Group Title

Thanks waterineyes.

3. klimenkov Group Title

$$(x-(a+bi))(x-(a-bi))=x^2-2ax+(a^2+b^2)$$ So, we get that this quadratic equation has a root $$x=a+bi$$. So as $$a$$ and $$b$$ are any real numbers we prove it for any complex number.

4. mathslover Group Title

Hmn wait let me check it .. :)

5. Mikael Group Title

What' remarkable in @klimenkov is that his coefficients are REAL

6. mathslover Group Title

Oh k so there is a mistake. That is : Where is quadratic equation @klimenkov ? it must have been : $\large{(X-(a+ib))(X-(a+ib))=\color{blue}{0}}$

7. mathslover Group Title

Ok I meant (a-ib) in one of them .. :)

8. UsukiDoll Group Title

ummm what subject is this?

9. klimenkov Group Title

1) Equation $$x^2-2ax+(a^2+b^2)$$ is QUADRATIC. 2) Equation $$x^2-2ax+(a^2+b^2)$$ has a root $$x=a+bi$$. 3) $$a+bi$$ is any complex numer for $$a\in \mathbb R, b \in \mathbb R$$.

10. klimenkov Group Title

Try to multiply carefully, there is no mistake.

11. mathslover Group Title

@klimenkov I was saying that, you didn't include 0 in RHS, why so?

12. klimenkov Group Title

Ah.. I'm really sorry. You are absolutely right. It was not an equation. It was just a polynomial.

13. mathslover Group Title

Oh no problem. That was a minor mistake ... No problem.. Thanks though.

14. waterineyes Group Title

@UsukiDoll this is Mathematics...

15. klimenkov Group Title

I like questions from people who have SmartScore > 75. Ask something interesting again!

16. mathslover Group Title

OK, if I will get a doubt, then I will like to ask you.

17. AravindG Group Title

am i late?

18. klimenkov Group Title

But if you have no need in real coefficients, you can just make an equation $$(x-(a+bi))^2=0$$

19. AravindG Group Title

@mathslover i would just observe the graph and stat this statement is true

20. AravindG Group Title

and would keep in mind that evry real number a can be written as a complex number a+0i

21. UsukiDoll Group Title

@wwaterineyes besides Mathematics. I meant what level of Mathematics is this because this certainly doesn't look like calculus or any lower level math question.

22. waterineyes Group Title

If you are asking for chapter then it is $$\mathsf{Complex \; \; Numbers...}$$

23. UsukiDoll Group Title

o_O

24. waterineyes Group Title

Ha ha ha ha...

25. waterineyes Group Title

And I was just kidding...

26. klimenkov Group Title

If you are asking for chapter then it is $$\text{Algebra}$$.

27. UsukiDoll Group Title

oh my O_O