# Find quadratic using its roots.

• January 27th 2010, 01:47 AM
carmo100
I was wondering how to go about doing this question:
Find all quadratics with the roots -3+-2i in the form ax^2 +bx+c=0

any help is appreciated.
• January 27th 2010, 01:54 AM
tonio
Quote:

Originally Posted by carmo100
I was wondering how to go about doing this question:
Find all quadratics with the roots -3+-2i in the form ax^2 +bx+c=0

any help is appreciated.

This is a nice, little theorem (a particular case of Viete's formulae):

Theorem: if $x_1\,,\,x_2$ are the roots of a quadratic equation $ax^2+bx+c=0$, then $x_1x_2=\frac{c}{a}\,,\,\,x_1+x_2=-\frac{b}{a}$

Tonio
• January 27th 2010, 02:01 AM
carmo100
Thankyou, this is exactly what I needed.
• January 27th 2010, 02:34 AM
HallsofIvy
Also, any quadratic having roots $r_1$ and $r_2$ is of the form $a(x- r_1)(x+ r_1)= 0$. Replace $r_1$ with -3+ 2i, $r_2$ with -3- 2i, and multiply that out.

The easiest way to do that is to think of it as a product of "sum and difference": ((x-3)- 2i)((x-3)+ 2i)