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

- mathslover

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

- chestercat

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- anonymous

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

- mathslover

Thanks waterineyes.

- 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.

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## More answers

- mathslover

Hmn wait let me check it .. :)

- anonymous

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

- mathslover

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

- mathslover

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

- UsukiDoll

ummm what subject is this?

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

- klimenkov

Try to multiply carefully, there is no mistake.

- mathslover

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

- klimenkov

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

- mathslover

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

- anonymous

@UsukiDoll this is Mathematics...

- klimenkov

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

- mathslover

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

- AravindG

am i late?

- klimenkov

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

- AravindG

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

- AravindG

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

- 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.

- anonymous

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

- UsukiDoll

o_O

- anonymous

Ha ha ha ha...

- anonymous

And I was just kidding...

- klimenkov

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

- UsukiDoll

oh my O_O

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