sums and products of roots

**pogiphilip** · September 2nd 2009, 12:58 AM

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

**red_dog** · September 2nd 2009, 01:25 AM

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.

**pogiphilip** · September 2nd 2009, 01:31 AM

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

**ynj** · September 2nd 2009, 01:40 AM

Originally Posted by pogiphilip

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

$(-m)^2-4\cdot 2=(a+(a+1))^2-4a(a+1)=(a-(a+1))^2=1$

**pogiphilip** · September 2nd 2009, 01:47 AM

i still dont understand what that is ynj

**ynj** · September 2nd 2009, 02:03 AM

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!

**pogiphilip** · September 2nd 2009, 02:08 AM

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

**ynj** · September 2nd 2009, 02:10 AM

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.

**pogiphilip** · September 2nd 2009, 02:40 AM

thank you for your help and i think i figured it out. thanks ynj and red_dog

**pogiphilip** · September 2nd 2009, 02:47 AM

also what are the roots if the two roots are consecutive?

**ynj** · September 2nd 2009, 09:32 AM

Originally Posted by pogiphilip

also what are the roots if the two roots are consecutive?

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

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