Question

Show that every complex number is a root of some quadratic equation.

Answer 1

I forgot to say :

Congratulations for \(\large \color{green}{\frak{green..}}\) @mathslover

Answer 2

Thanks waterineyes.

Answer 3

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

Answer 4

Hmn wait let me check it .. :)

Answer 5

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

Answer 6

Oh k so there is a mistake. That is :

Where is quadratic equation @klimenkov ?

it must have been :
\[\large{(X-(a+ib))(X-(a+ib))=\color{blue}{0}}\]

Answer 7

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

Answer 8

ummm what subject is this?

Answer 9

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 \(a\in \mathbb R, b \in \mathbb R\).

Answer 10

Try to multiply carefully, there is no mistake.

Answer 11

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

Answer 12

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

Answer 13

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

Thanks though.

Answer 14

@UsukiDoll this is Mathematics...

Answer 15

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

Answer 16

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

Answer 17

am i late?

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

@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.

Answer 22

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

Answer 23

o_O

Answer 24

Ha ha ha ha...

Answer 25

And I was just kidding...

Answer 26

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

Answer 27

oh my O_O