Thread: quadratics and imaginary roots

Hi;
If a quadratic polynomial won't factor does this mean it
has imaginary roots?

if I complete the square on the same polynomial which makes it
factorable does this process give it real roots?

Originally Posted by anthonye

Hi;
If a quadratic polynomial won't factor does this mean it
has imaginary roots?
if I complete the square on the same polynomial which makes it
factorable does this process give it real roots?

Consider , is that what you mean?

Yes

Originally Posted by anthonye

Hi;
If a quadratic polynomial won't factor does this mean it
has imaginary roots?

If it won't factor over the reals, then yes, it will have COMPLEX roots (which have a real part and an imaginary part).

if I complete the square on the same polynomial which makes it
factorable does this process give it real roots?

Completing the square won't magically make a quadratic factorise over the reals. You WILL however be able to factorise over the complex numbers, which will still give complex roots.

Ok thank you Plato and prove it.

Originally Posted by anthonye

Yes

The point is that any quadratic with real coefficients can be factored.
Take the example I gave you. has two real roots: .

The factored form is: .

On the other hand, has no real zeros but also can be factored: (x- i)(x+ i). The trouble is, you didn't say what you meant by "factored".

Please can you elaborate on the fact that any quadratic with real coefficients can be factored

I'm a little lost with that.

Any quadratic, whether its coefficients are real or not, can be factored- but NOT necessarily with integer or even real coefficients.

Take as a generic example. First, factor out the "a". . Now "complete the square"- add and subtract : . The point of that is that , a perfect square. And of course, .

That is, we can write

And now we use the fact that to write


That is factored into a number times two linear factors. With those square roots, those numbers may not be integers or even real numbers.

Wow ok thanks Hallsofivy Ineed to look at this more any idea where I can read about this subject?