Thread: sum and product of roots

1. sum and product of roots

I have just been reading about the relationship between the roots of a quadractic and the coefficents.

The book used identities to show if x , y are the roots of the equation aqz^2+bz+c=0, then -b/a=x+y and c/a=xy. Then it asked me to find the equation with roots whose sum and product were known.

I was doing the questions but I was thinking surely something else needs to be true. Am I correct in saying in order to do this question, you need to know that the converse is true as well. I.E. If x+y =-b/a and xy=c/a, then x,y are the roots of the equation. This can be seen by solving the relations simultaneously and finding that the only 2 x,y satisfying the relations are the two roots of the quadractic equation.

2. Re: sum and product of roots

Originally Posted by Duke
The book used identities to show if x , y are the roots of the equation az^2+bz+c=0, then -b/a=x+y and c/a=xy. Then it asked me to find the equation with roots whose sum and product were known.
The roots of $\displaystyle az^2+bz+c=0$ are the roots of $\displaystyle z^2+\frac{b}{a}z+\frac{c}{a}=0$ (of course $\displaystyle a\ne 0$).

But by the factor theorem
$\displaystyle z^2+\frac{b}{a}z+\frac{c}{a}=(z-x)(z-y)=z^2-(x+y)z+(xy).$

3. Re: sum and product of roots

Hello, Duke!

Your use of $\displaystyle x$ and $\displaystyle y$ is confusing.
I'll modify it to "standard" notation.

$\displaystyle \text{I have just been reading about the relationship}$
. . $\displaystyle \text{between the roots of a quadractic and the coefficents.}$

$\displaystyle \text{The book used identities to show:}$
. . $\displaystyle \text{If }p, q\text{ are the roots of the equation }ax^2+bx+c\:=\:0,$
. . $\displaystyle \text{ then: }\:p+q \,=\,\text{-}\tfrac{b}{a}\,\text{ and }\,pq \,=\,\tfrac{c}{a}$

$\displaystyle \text{Then it asked me to find the equation with roots}$
. . $\displaystyle \text{whose sum and product were known.}$

$\displaystyle \text{I was doing the questions, but I was thinking:}$
. . $\displaystyle \text}surely something else needs to be true.}$

$\displaystyle \text{Am I correct in saying in order to do this question,}$
. . $\displaystyle \text{you need to know that the converse is true as well?}$

$\displaystyle \text{That is: \:if }p+q \,=\,\text{-}\tfrac{b}{a}\,\text{ and }\,pq\,=\,\tfrac{c}{a},$
. . $\displaystyle \text{ then }p,q\text{ are the roots of the equation.}$

Good thinking!

Yes, you are correct.

Those roots/coefficients relations are usually given
. . (or should be given) with an "if and only if".