Thread: sums and products of roots

1. 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

2. 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.

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

4. Originally Posted by pogiphilip
thanks for your help but i havent leant that so could you explain to me how that works?
$(-m)^2-4\cdot 2=(a+(a+1))^2-4a(a+1)=(a-(a+1))^2=1$

5. i still dont understand what that is ynj

6. 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!

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

8. 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.

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

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

11. Originally Posted by pogiphilip
also what are the roots if the two roots are consecutive?
you may use formula to solve it .......