# sums and products of roots

• Sep 2nd 2009, 12:58 AM
pogiphilip
sums and products of roots
i dont understand this question:
in the quadratic equation x²+mx+2=0, the roots are consecutive. find the values of m.
i dont understand the part of the consecutive roots
• Sep 2nd 2009, 01:25 AM
red_dog
The roots are $a$ and $a+1, \ a\in\mathbb{Z}$

Then $\left\{\begin{array}{ll}a+a+1=-m\\a(a+1)=2\end{array}\right.$

Solve the second equation, find a, then replace a in the first equation and find m.
• Sep 2nd 2009, 01:31 AM
pogiphilip
thanks for your help but i havent leant that so could you explain to me how that works?
• Sep 2nd 2009, 01:40 AM
ynj
Quote:

Originally Posted by pogiphilip
thanks for your help but i havent leant that so could you explain to me how that works?

$(-m)^2-4\cdot 2=(a+(a+1))^2-4a(a+1)=(a-(a+1))^2=1$
• Sep 2nd 2009, 01:47 AM
pogiphilip
i still dont understand what that is ynj
• Sep 2nd 2009, 02:03 AM
ynj
Quote:

Originally Posted by pogiphilip
i still dont understand what that is ynj

Do you know Vieta's Theorem?
if a quadratic equation $ax^2+bx+c=0$ have two roots $\alpha,\beta$, then $a(x-\alpha)(x-\beta)=ax^2+bx+c\Leftrightarrow ax^2-a(\alpha+\beta)x+a\alpha\beta=0$
Thus $\alpha+\beta=-\frac{b}{a},\alpha\beta=\frac{c}{a}$to keep all the coefficient identical.
So $a+(a+1)=-m,a(a+1)=2$, $(-m)^2-4\cdot 2=(a+(a+1))^2-4a(a+1)=(a-(a+1))^2=1$, Then you can solve m!
• Sep 2nd 2009, 02:08 AM
pogiphilip
but also the question says to use roots that are consecutive?, and i don't know vieta's theorem.
• Sep 2nd 2009, 02:10 AM
ynj
Quote:

Originally Posted by pogiphilip
but also the question says to use roots that are consecutive?, and i don't know vieta's theorem.

That means two roots are in the form a,a+1. Vieta's Theorem has been proved in my previous post.
• Sep 2nd 2009, 02:40 AM
pogiphilip
thank you for your help and i think i figured it out. thanks ynj and red_dog
• Sep 2nd 2009, 02:47 AM
pogiphilip
also what are the roots if the two roots are consecutive?
• Sep 2nd 2009, 09:32 AM
ynj
Quote:

Originally Posted by pogiphilip
also what are the roots if the two roots are consecutive?

you may use formula to solve it .......