1. ## 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?

2. ## Re: quadratics and imaginary 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 $3x^2-3x-2$, is that what you mean?

Yes

4. ## Re: quadratics and imaginary roots

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.

5. ## Re: quadratics and imaginary roots

Ok thank you Plato and prove it.

6. ## Re: quadratics and imaginary roots

Originally Posted by anthonye
Yes
The point is that any quadratic with real coefficients can be factored.
Take the example I gave you. $3x^2-3x-2=0$ has two real roots: $\left( {\frac{{3 + \sqrt {33} }}{6}} \right)\;\& \;\left( {\frac{{3 - \sqrt {33} }}{6}} \right)$.

The factored form is: $3x^2-3x-2=\left[ {x - \left( {\frac{{3 + \sqrt {33} }}{6}} \right)} \right]\left[ {x - \left( {\frac{{3 - \sqrt {33} }}{6}} \right)} \right]=0$.

7. ## Re: quadratics and imaginary roots

On the other hand, $x^2+1$ 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".

8. ## Re: quadratics and imaginary roots

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

I'm a little lost with that.

9. ## Re: quadratics and imaginary roots

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

Take $ax^2+ bx+ c$ as a generic example. First, factor out the "a". $a(x^2+ \frac{b}{a}x+ \frac{c}{a})$. Now "complete the square"- add and subtract $\frac{b^2}{4a^2}$: $a(x^2+ \frac{b}{a}x+ \frac{b^2}{4a}+ \frac{c}{a}- \frac{b^2}{4a^2})$. The point of that is that $x^2+ \frac{b}{a}x+ \frac{b^2}{4a^2}= (x+ \frac{b}{2a})^2$, a perfect square. And of course, $\frac{b^2}{4ac^2}-\frac{c}{a}= \left(\sqrt{\frac{b^2}{4a^2}- \frac{c}{a}}\right)^2$.

That is, we can write $ax^2+ bx+ c= a\left(\left(x+ \frac{b}{2a}\right)^2- \left(\sqrt{\frac{b^2}{4a^2}- \frac{c}{a}}\right)^2\right)$

And now we use the fact that $x^2- a^2= (x- a)(x+ a)$ to write
$a\left(x+ \frac{b}{2a}+\sqrt{\frac{b^2}{4a^2}- \frac{c}{a}}\right)\left(x+ \frac{b}{2a}- \sqrt{\frac{b^2}{4a^2}- \frac{c}{a}}\right)$

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