## mathslover 3 years ago Show that every complex number is a root of some quadratic equation.

1. waterineyes

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

2. mathslover

Thanks waterineyes.

3. klimenkov

$$(x-(a+bi))(x-(a-bi))=x^2-2ax+(a^2+b^2)$$ 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

Hmn wait let me check it .. :)

5. Mikael

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

6. mathslover

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

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

8. UsukiDoll

ummm what subject is this?

9. klimenkov

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

Try to multiply carefully, there is no mistake.

11. mathslover

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

12. klimenkov

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

13. mathslover

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

14. waterineyes

@UsukiDoll this is Mathematics...

15. klimenkov

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

16. mathslover

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

17. AravindG

am i late?

18. klimenkov

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

19. AravindG

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

20. AravindG

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

21. UsukiDoll

@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

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

23. UsukiDoll

o_O

24. waterineyes

Ha ha ha ha...

25. waterineyes

And I was just kidding...

26. klimenkov

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

27. UsukiDoll

oh my O_O